Interest Rate Modeling For Risk Management [2nd ed.] 1681086891, 9781681086897

Interest Rate Modeling for Risk Management presents an economic model which can be used to compare interest rate and per

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Economics: Current and Future Developments Volume 1 6HFRQG(GLWLRQ ,QWHUHVW5DWH0RGHOLQJIRU5LVN0DQDJHPHQW 0DUNHW3ULFHRI,QWHUHVW5DWH5LVN

Authored by Takashi Yasuoka Graduate School of Engineering Management Shibaura Institute of Technology 3-9-14 Shibaura, Minato-ku, Tokyo, Japan



(FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV Volume # 1 QG(GLWLRQ ,QWHUHVW5DWH0RGHOLQJIRU5LVN0DQDJHPHQW 0DUNHW3ULFHRI,QWHUHVW5DWH5LVN $XWKRU: 7DNDVKL 0. That is, Ti+1 − Ti = δ for all i < n. Note that the floating payment δL(i) takes place at time Ti+1 , and so its present value is given by L(i)D(i + 1). Since the present value of the fixed cash flows should be equal to that of the floating cash flows, we obtain S(n)

n−1 

δD(i + 1) =

n−1 

δD(i + 1)L(i).

(1.15)

i=0

i=0

From (1.12), it holds that

δD(i + 1)L(i) = D(i) − D(i + 1).

(1.16)

Taking the sum of (1.16) over i = 0, · · · , n − 1, the right side of (1.15) becomes δ

n−1  i=0

D(i + 1)L(i) =

n−1  i=0

D(i) − D(i + 1)

= 1 − D(n).

(1.17)

(FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G 7

Interest Rate Risk

Substituting the above into (1.15), the swap rate can be represented by the discount factors as 1 − D(n) S(n) = n−1 . (1.18) i=1 δD(i + 1) Forward rate building From (1.18), we have a system of linear equations such that m−1  δD(i + 1) + D(m) 1 = S(m) = S(m)

i=1 m−2 

δD(i + 1) + (1 + δS(m))D(m),

(1.19)

i=1

for m = 1, · · · , n/δ. Alternatively, this can be represented by the following equation. ⎞ ⎛ ⎞ ⎞⎛ ⎛ 1 + δS(1) 0 ··· 0 1 D(1) .. ... ⎟ ⎜ D(2) ⎟ ⎜ 1 ⎟ ⎜ 1 + δS(2) . ⎟ ⎜ ⎟ ⎟⎜ ⎜ δS(2) ⎟ ⎜ .. ⎟ = ⎜ .. ⎟ (1.20) ⎜ . . .. .. .. ⎝ ⎠ ⎝ . ⎠ ⎝ . ⎠ . 0 D(n) 1 δS(n) δS(n) · · · 1 + δS(n) Then, from all swap rates S(m), m = 1, · · · , n, we can obtain all discount factors D(m), m = 1, · · · , n by solving the above equation. Fortunately, the coefficient matrix of (1.20) is lower-triangle, and so the above equation is readily solved by forward substitution as follows. 1) D(1) is directly solved from the first row in (1.20) by D(1) = 1/(1 + δS(1)). 2) Substituting D(1) into the second equation, we have (1 + δS(2))D(2) = 1 − δS(2)D(1).

Solving D(2) from this equation, it follows that 1 − δS(2)D(1) D(2) = . 1 + δS(2)

(1.21)

(1.22)

8 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

Takashi Yasuoka

3) Substituting D(1) and D(2) into the third equation, we obtain (1 + δS(3))D(3) = 1 − δS(3)(D(1) + D(2)).

(1.23)

Then, 1 − δS(3)(D(1) + D(2)) . 1 + δS(3) 4) Inductively, we have m−1 1 − δS(m)( i=1 D(i)) D(m) = . 1 + δS(m) D(3) =

(1.24)

(1.25)

for all m ≤ n. This algorithm is known as bootstrapping (cf. Nawalkha et al. (2005)). The discount factors are computed with respect to all m by solving the above equation. Finally, the forward LIBOR is determined from the discount factors by using (1.16) as L(i) =

D(i) − D(i + 1) δD(i + 1)

(1.26)

for all i. We remark that it is not possible to observe all swap rates for S(m), m = 1, · · · , n in an actual market. In the practice, some values of S(m) are computed by interpolation. 1.3

Term Structure of Interest Rates

The yield to maturity of bonds can be observed in the bond market. The yield curve is a graph that plots time to maturity on the horizontal axis and the yield to maturity on the vertical axis, which visually exhibits the term structure of interest rates. The swap rate curve and the forward LIBOR curve are similarly exhibited. For example, from (1.15) the swap rate S(n) is represented by a series of forward LIBORs L(i), i = 1, · · · , n − 1 as n−1 i=0 δD(i + 1)L(i) S(n) =  . (1.27) n−1 i=0 δD(i + 1)

(FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G 9

Interest Rate Risk %

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

1

2

3

4

5

6

7

8

9

Term to maturity (year) (a) Yield curve %

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0

1

2

3

4

5

6

7

8

9

Maturity (year) (b) Six-month implied forward rate

Figure 1.2:

Yield curve and the implied forward rate. U.S. Treasury market on 25 July 2014. Yield data were retrieved from the Bloomberg site. The author calculated the implied forward rates and created the figure.

This means that the swap rate consists of forward LIBORs L(i) for i < n. This property is the same when considering the yield to maturity. In contrast, the forward rate is the obtained interest rate of the corresponding future period. Therefore, the forward rate curve shows us the term structure of interest rates more clearly than the yield curve and the swap rate curve can. For this reason, the forward rate curve is often used to analyze the interest rate market. In particular, the forward rates derived from a term structure of interest rates are called the implied forward rate. For example,

10 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

Takashi Yasuoka

the “implied” forward rate can be practically observed in a government bond market or a swap market. Example 1.3.1: Yield curve and forward rate curve Fig. 1.2(a) exhibits the yield curve for U.S. Treasury bills on 25 July 2014. From this data, the six-month forward rates are implied; these are shown in Fig. 1.2(b). The graph shows that the forward rate curve dips at around five years. That is, the forward rates for the periods [5, 5.5] and [5.5, 6] (years) are relatively lower than those of other periods. This is typical of how the forward rate curve is usually observed by market participants, including bond investors and derivatives traders. Example 1.3.2: Historical data on interest rates Market interest rates are continuously fluctuating. As an example, Fig. 1.3 shows a monthly historical chart of six-month forward rates in the U.S. Treasury market. This chart suggests that movements in forward rates are uncertain and unpredictable. Therefore, the evolution of the forward rates can reasonably be mathematically modeled as a stochastic process. For this, the term structure model of interest rate is constructed so that it simulates the forward-rate dynamics, which will be introduced in Chapters 3 and 5. 1.4

Interest Rate Risk of Bonds

This section summarizes a basic risk measure for the interest rate risk of bonds. First, we examine the relation between bond price and bond yield by calculating some simple examples. Then, we introduce sensitivity risk and convexity risk by referring to the example calculations. Bond price and yield Let us consider an n-year bond with annual coupon rate c. We denote by r the annual yield to maturity at n years, which is supposed to be the market yield. This yield is used as the discount rate, and the present price p of the bond is obtained from (1.6) as p=

c 1+c c + + ··· + . 2 (1 + r) (1 + r) (1 + r)n

(1.28)

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Interest Rate Risk

0.07

spot 1 2 3 4 5 6 7 8 9

0.06 0.05 0.04 0.03 0.02 0.01 Jul-12

Oct-12

Jan-12

Apr-12

Jul-11

Oct-11

Jan-11

Apr-11

Jul-10

Oct-10

Apr-10

Jan-10

Jul-09

Oct-09

Apr-09

Jan-09

Jul-08

Oct-08

Apr-08

Jan-08

Oct-07

0

Figure 1.3:

Historical chart of implied forward rates. U.S. Treasury market from Oct 2007 to Oct 2012. In the graph, a number such as “3” indicates the forward rate over the period [3, 3.5] (years). Yield data were retrieved from the Board of Governors of the Federal Reserve System (http://www.federalreserve.gov/releases/h15/data.htm) Implied forward rates were calculated by the author.

Example 1.4.1 As an example, let us consider a five-year bond with annual coupon rate of 5%. From (1.28), the present price of this bond is calculated as 0.05 0.05 0.05 1.05 0.05 + + + + 2 3 4 1.05 (1.05) (1.05) (1.05) (1.05)5 0.05 0.05 0.05 0.05 1.05 + + + + = 1.05 1.1025 1.157625 1.215506 1.276282 = 0.04762 + 0.04535 + 0.04319 + 0.04114 + 0.82270

p =

= 1.00.

(1.29)

Thus, we can buy this bond at face value. Next, we consider a three-year bond with annual coupon rate of 5%. The

12 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

Takashi Yasuoka

price is obtained by 0.05 0.05 1.05 + + 2 1.05 (1.05) (1.05)3 0.05 1.05 0.05 + + = 1.05 1.1025 1.157625 = 0.04762 + 0.04535 + 0.90703

p =

= 1.00.

(1.30)

As with the five-year bond, we can buy the three-year bond at face value. Generally, the bond price is par (i.e., face value) when the market yield coincides with the coupon rate. Interest rate sensitivity We define the interest rate sensitivity of bonds with respect to yield by ∂p c 2c n(1 + c) − − · · · − < 0. =− ∂r (1 + r)2 (1 + r)3 (1 + r)n+1

(1.31)

This represents the degree of change in the price of bonds in response to fluctuations in market interest rates. For business, the basis point value (BPV) corresponds to the interest rate sensitivity. Traditionally, duration is also used as a measure of interest rate sensitivity. From the right side of (1.31), it obviously holds that ∂p/∂r < 0. Thus, interest rate sensitivity is always negative, that is, a rise in interest rates will cause bond prices to fall. If we assume that c ≈ 0 and r ≈ 0, then we have the approximation ∂p ≈ −n. ∂r

(1.32)

Thus, bonds with longer time to maturity show larger changes in price than those with shorter time to maturity. This suggest that the longer a bond has until maturity, the larger the interest rate risk is. Example 1.4.2: Interest rate sensitivity Consider the three-year bond in Example 1.4.1. The interest rate sensitivity

Interest Rate Risk

(FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G 13

is calculated by (1.31) as follows. ∂p 0.05 2 × 0.05 3 × 1.05 = − − − 2 ∂y (1.05) (1.05)3 (1.05)4 0.1 3.15 0.05 − − = − 1.1025 1.157625 1.215520 = −0.04535 − 0.08639 − 2.59151 = −2.72325

(1.33)

From this, (1.32) implies that ∂p/∂r ≈ −3. Hence, we may consider that (1.32) roughly approximates ∂p/∂r. Convexity If the three-year yield falls to 4%, then the price of the three-year bond will become 0.05 1.05 0.05 + + 2 1.04 (1.04) (1.04)3 0.05 1.05 0.05 + + = 1.04 1.0816 1.124864 = 0.04808 + 0.04623 + 0.93344

p =

= 1.02775.

(1.34)

Then, a 1% yield rise causes a 2.775% price fall, which is somewhat different from the result of (1.33). The difference here is caused by interest rate sensitivity that is not constant with respect to the level of the interest rate. Indeed, (1.31) is a nonlinear function of r. The second derivative of the price p is represented by ∂ 2p 2c 6c n(n − 1)c = + + ··· + . 2 3 4 ∂r (1 + r) (1 + r) (1 + r)n+2

(1.35)

We see that the right hand is always positive, and from this the price function of (1.31) is strictly convex. If, instead, the three-year yield rises to 6%, then the price of the above

14 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

Takashi Yasuoka

three-year bond will be 0.05 1.05 0.05 + + p = 2 1.06 (1.06) (1.06)3 0.05 0.05 1.05 + + = 1.06 1.1236 1.191016 = 0.04717 + 0.04450 + 0.88160 = 0.97327.

(1.36)

Here, a price fall of 2.673% results in a 1% yield rise. The percentage increase in price is greater than the percentage decrease in price. This difference represents the degree of convexity of (1.28). Convexity risk Assume that we hold a three-year bond when the three-year yield is 5%. Let us consider the interest rate future, whose interest rate sensitivity is 1% to a 1% change in three-year yield. Here, we may assume that the interest rate future has convexity of 0. If we sell the future for 2.723 units, then our interest rate risk becomes 0 today. If the yield falls to 4%, then we lose 2.775% from the bond and gain 2.723% from the future. We lose from this trade, by a hedge error of −2.775 + 2.723 = −0.052 (%). This hedge error represents the risk caused by convexity, which is referred to as convexity risk. Fig. 1.4 contains graphs of the bond price relative to the yield for three-, five-, and ten-year bonds with a coupon rate of 5%. This shows that the slope of the curve is larger for longer maturity bonds, and all the curves seem to be slightly convex. Remark about interest rate swaps Sensitivity is used as a measure of risk for interest rate swaps. The mechanism is the same for bond sensitivity, with the sensitivity of the swap being higher for longer terms to maturity and lower for shorter terms to maturity. Naturally, the swap value also has the convexity property. 1.5

Value at Risk

This section studies risk measures for portfolios. For the sake of simplifying descriptions, we assume that the portfolio has only assets, with liabilities re-

(FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G 15

Interest Rate Risk

Price 1.2 1.1 1 0.9 3

0.8

5

0.7

10 0.6 3

4

5

6

7

Yield (%)

Figure 1.4: Bond price to yield for 3-, 5-, and 10-year bonds. garded as negative assets. We additionally regard a single asset as a portfolio. There are two kinds of risk measures to assess the portfolio risk. One is sensitivity, which is the price change in a portfolio resulting from changes in the risk factor. Another is VaR. We begin by addressing the risk factors. Risk factors of portfolio The value of a portfolio is affected by changes in risk factors. The basic risk factors for financial assets consist of market risk, credit risk, liquidity risk, and so on. The market risk is generally classified into interest rate risk, exchange rate risk (also called currency risk), and stock price risk. The interest rate risk of bonds was discussed in Section 1.4 of this chapter. Credit risk is the risk of default by the counterparty, such as the risk that a bond issuer will default on payments. Liquidity risk is the risk that arises from the difficulty of selling an asset; sometimes it is reflected in widening of the bid–ask spread. ⎛

market risk (market risk, exchange rate, stock price) ⎜ credit risk risk factors = ⎜ ⎝ liquidity risk etc. (1.37)

16 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

Takashi Yasuoka

Basically, the risk factor of government bonds is the interest rate risk only. The risk factors of foreign government bonds are exchange rate risk and foreign interest rate risk. Sometimes, sovereign credit risk is non-negligible. Risk measures: Sensitivity Let x1 , · · · denote factors with risk, such as yield, forward rate, and exchange rate. We assume that the price of a portfolio is represented by p(x1 , x2 , · · · ). Then, the xi -sensitivity of a portfolio is defined by ∂p(xi )/∂xi , for i = 1, 2, · · · . For example, the bond price p is represented in (1.28) by a function of yield r, as 1+c c c p= + + · · · + . (1.38) (1 + r) (1 + r)2 (1 + r)n The interest rate sensitivity is given in (1.31) by the partial derivative with respect to r, as ∂p 2c n(1 + c) c =− − − ··· − . (1.39) 2 3 ∂r (1 + r) (1 + r) (1 + r)n+1 In practice, there are several measures of interest rate sensitivity. The basis point value (BPV) is the sensitivity to a change of 0.01% (= 1 basis point) in the forward rates. The “delta” of an option is that option’s sensitivity to the underlying risk factor. These sensitivities are significant measures for both assessing the risk of the portfolio and determining the hedge ratio of the portfolio. For details, see, among others, Hull (2000) and Sadr (2009). Linearity of portfolio If the value of a portfolio is represented by a linear combination of risk factors, such that p(x1 , x2 , · · · ) = c0 + c1 x1 + c2 x2 + · · · ,

(1.40)

then the portfolio is called linear to x1 , x2 , · · · . For example, consider a forward contract of some asset with a delivery date of T . We denote by r the simple interest rate to T and by p the present price

Interest Rate Risk

(FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G 17

Cap price

In the money At the money Out of the money K

Forward LIBOR

Figure 1.5: Convex property of cap price for the forward LIBOR of the asset. Then, the forward price is given by Fp = p/(1 + rT ). Therefore, when regarding the asset price as a risk factor, Fp is linear to p. Similarly, regarding the underlying asset as a risk factor for futures implies that the future price is also linear to p. As an example, interest rate futures can be considered as linear to the forward LIBOR. Convexity It is more usual for a portfolio to be nonlinear. Indeed, we have already seen that the price of fixed-coupon bonds is not linear, as shown in (1.39). Furthermore, a call/put option written on some underlying risk factor has noteworthy nonlinearity. For example, let us consider a European (interest rate) cap with strike rate K and some expiry date T > 0. The price of the cap at t = 0 is conceptually illustrated in Fig. 1.5. The price is low and flat out of the money, but the slope of the price curve becomes steeper in the money. The convexity is at maximum near the point where the price is at the money. Thus, European cap/floor options written on the forward LIBOR are the most basic example that exhibits such convexity. Additionally, it is well known that structured products involving derivatives exhibit strong convexity. Therefore, sensitivity is not sufficient to measure the risk from large changes in risk factors. Value at risk

18 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

Takashi Yasuoka

In financial risk management, “risk” means the downside risk of the portfolio, rather than sensitivity and convexity. In this context, VaR is a standard measure for financial risk management. Briefly, VaR is the maximum expected loss over a given holding period, calculated at a certain confidence level. Next, we give the quantitative definition of VaR. Let pt denote the value of a portfolio at time t, with p0 as the current value of the portfolio. The value of pt moves uncertainly according to time. The loss of this portfolio over a holding period T is given by a random variable L, where L = p0 − pT , with a loss represented by a positive value. To measure the downside risk, we build the profit–loss (P&L) distribution of the portfolio at time T from historical data, as shown in Fig. 1.6. Let α be a constant with 0 < α < 1. Usually, we set α as a small value, taking, for example, α = 0.01 or 0.05. The VaR at confidence level of 1 − α over the holding period T is defined by the 100(1 − α)th centile, such that V aRα = − inf{l : P(L ≤ l) > α},

(1.41)

where P denotes the probability of the event L ≤ l, as given by the P&L distribution. In practice, V aRα is referred to as 100(1 − α)%V aR. For example, 99%V aR denotes V aR0.01 . Meaning of VaR The definition of VaR formalizes the potential loss in value of the portfolio over a holding period T for a given confidence level 1 − α. For example, Fig. 1.6 illustrates that the 95% confidence level (95th centile) shows the loss of 1 million U.S. dollars over one month. For that case, the 1-month VaR of the portfolio is estimated as 1 million dollars; that is, there is a 5% chance that the loss in value will be more than 1 million dollars in any given month. As is customary in the literature, VaR is expressed as a positive value in this book. 1.6

Computing VaR

It is possible to theoretically evaluate sensitivity and convexity, which are deterministic, but not to evaluate VaR, because it contains stochastic characteristics. Three basic types of models are used to measure VaR:

(FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G 19

Interest Rate Risk

5%

Time

Distribution of P&L at t=1 month 95%

1 month Profit

Loss

$1 M Asset value at t=0

t=0 Value

Figure 1.6: VaR and the distribution of asset value 1) covariance VaR models; 2) historical simulation models; and 3) Monte Carlo simulation models. We briefly explain the concepts of these models below, and additionally introduce nested simulation for use with derivative portfolios. 1.6.1

Covariance VaR Models

One-factor model The covariance VaR model assumes that the portfolio value is linear to the risk factor and normally distributed. Then, the key parameters are the mean and variance of the P&L distribution, which are measured from historical data. Let us consider a portfolio with a single risk factor x, the value of which is denoted by p(x). From the linearity assumption, we can model p(x) as

20 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

Takashi Yasuoka

p(x) = a+bx for some positive constants a and b. Let μ and σ 2 be, respectively, the mean and variance of x. For convenience, we assume that p(μ) is equal to the current value of the portfolio. It is known from the normality of the distribution that the 95% confidence level of x is μ − 1.645σ, and the 99% confidence level is μ − 2.326σ. From this, we can see that the 95%VaR is given by 95%V aR = {a + μb} − {a + b(μ − 1.645σ)} = 1.645σb.

(1.42)

Fig. 1.7 may help in understanding this scheme. Similarly, we have 99%V aR = 2.326σb.

(1.43)

When p(x) is not linear but almost linear, that is, when p(x) ≈ a + bx, we can approximate the VaR from (1.42) and (1.43). Multi-factor model Generally, portfolios have several risk factors. For a model with multiple risk factors, the covariance VaR is estimated by combining the individual means and variances of factors with the covariances among sets of factors. Let us consider a two-factor model. We assume that the value of a portfolio is represented by two risk factors, x1 and x2 , and that this portfolio is linear to x1 and to x2 . Then, for some positive constants a, b1 , and b2 , we have p(x) = a + b1 x1 + b2 x2 .

(1.44)

Let μi and σi2 be the mean and variance of factor xi , with i = 1, 2. We denote by ρ the correlation coefficient between x1 and x2 . From the assumption that x1 and x2 are normally distributed, a + b1 x1 + b2 x2

(1.45)

follows a normal distribution with mean a + μ1 + μ2 and variance σ 2 , where σ 2 = b21 σ12 + b22 σ22 + 2b1 b2 ρσ1 σ2 .

(1.46a)

Equivalently, σ 2 = b21 σ12 + b22 σ22 + 2Cov(x1 , x2 ),

(1.46b)

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Interest Rate Risk

Profit p=a+bx Distribution of p

95%VaR

99%VaR Distribution of x Loss

μ-1.645σ

μ

x

μ-2.326σ

Figure 1.7: Covariance VaR in a one-factor model since Cov(x1 , x2 ) = ρσ1 σ2 . We assume that the current value of the portfolio is equal to p(μ1 , μ2 ). As in the one-factor model, the 95%VaR is obtained by

95%V aR = {a + b1 μ1 + b2 μ2 } − {a + b1 μ1 + b2 μ2 − 1.645σ} = 1.645σ.

(1.47)

Similarly, we have 99%V aR = 2.326σ. The advantage of the covariance VaR is that it is simple and quick to calculate. The disadvantage is that the assumption of linearity restricts the portfolios to which it can be applied. Moreover, the covariance is known to be difficult to estimate because it depends strongly on the sampled period.

22 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

1.6.2

Takashi Yasuoka

Historical Simulation Models

In historical simulation models, the empirical distribution of the P&L is directly constructed from historical data. For simplicity, we consider a portfolio consisting of two assets, A and B. For a holding period ΔT , let t1 , · · · , tJ+1 be a sequence of past days with ti − ti+1 = ΔT and t1 = 0. We let pA (i) and pB (i) denote the prices of A and B, respectively, at ti . Let rA (i) and rB (i) be the rates of return for A and B, respectively. These are defined by rA (i) =

pA (i) − pA (i + 1) pA (i + 1)

(1.48a)

rB (i) =

pB (i) − pB (i + 1) pB (i + 1)

(1.48b)

for i = 1, · · · , J. For the present values pA (0) and pB (0), the P&L distribution of the portfolio is built from the set {pA (0)rA (i) + pB (0)rB (i)}i=1,··· ,J ,

(1.49)

as shown in Fig. 1.6. From this distribution, we obtain the VaR, with the process working according to the idea shown in Fig. 1.6. Even for a very large portfolio, we can measure the VaR in the same way. It is a point in favor of this model that it does not require assumptions about neither the distribution of prices nor the linearity of assets. However, it is difficult to collect the sufficient historical data for this method. Even when there are sufficiently many data points from a long period of observation, some data might be too old to reasonably reflect future risk, which is needed for measuring VaR. 1.6.3

Monte Carlo Simulation Models

A Monte Carlo method is a simulation technique that uses randomly generated numbers for simulation. Here, distribution functions of risk factors are created by using a sequence of random numbers.

Interest Rate Risk

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Let T be a holding period. For simplicity, we assume a single risk factor and let p(x) and x denote, respectively, the value of the portfolio and the value of the risk factor at T . We assume that x is normally distributed with mean μ and variance σ 2 , and that the current value of the portfolio is equal to p(μ). Let zi , i = 1, · · · , J be a sequence of numbers generated randomly according to a standard normal distribution, where J is the number of simulation runs. Then, we can assign xi = μ + σzi . By regarding generated sequences xi , i = 1, · · · , J as scenarios for the risk factor x, we can obtain a distribution p(xi ), where the probability of each scenario is 1/J. Next we rearrange {p(xi )}i=1,··· ,J to {qk }k=1,··· ,J such that {p(xi ); i = 1, · · · , J} = {qk ; k = 1, · · · , J}

(1.50)

with qk ≤ qk+1 for all k. To obtain a confidence level 1 − α, we set kα = αJ. Then, (1 − α)100%VaR is given by −1 (qkα + qkα +1 ). (1.51) V aRα = 2 It is advantageous that Monte Carlo simulation models are applicable to nonlinear assets and path-dependent assets. This makes it possible to consider a great many scenarios. Moreover, we can employ this technique to the arbitrage-free model, which we will introduce in Chapter 3. The disadvantages of Monte Carlo simulation are high computational time and increased model risk. For a more advanced treatment of Monte Carlo simulation in financial engineering, interested readers are recommended to consult Glasserman (2004). Remark Monte Carlo simulation models should be built under the real-world measure (for definitioin, see Section 2.1.), but it is not a trivial exercise to use the arbitrage-free model under that measure. The reasons for this difficulty are addressed in Section 4.2. A key purpose of this book is to study the theory of arbitrage-free models under the real-world measure, and we will work with these in the later part of this book.

24 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

x

Outer step

Takashi Yasuoka

Inner step xT

pT¦xT x0

0

T

Figure 1.8: Nested simulation 1.6.4

Nested Simulation

There are many exotic options whose price depends on the path of an underlying asset. If a structured product involves an exotic derivative, then the price of the product is also path-dependent. Since the pricing of complex products is not trivial, recombining trees and Monte Carlo simulation are employed for pricing. When a portfolio contains exotic products such as those described above, it is difficult to price them for a future date, which is necessary for measuring VaR. Nested simulation is a computing technique for measuring VaR of a derivatives portfolio, which we briefly introduce below. Let T be a holding period for measuring VaR. We denote by xt the value of the risk factor at time t, and by pT the price of a derivative asset at T . As illustrated in Fig. 1.8, the distribution of the risk factor xT at T is simulated by a Monte Carlo method in the outer step. Then, we consider xT to represent a scenario in which VaR is to be measured. At each scenario xT , the derivative is priced as pT |xT by another Monte Carlo simulation starting at time T . This is the inner step. Once we obtain the distribution of pT |xT , we can compute

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the VaR in the manner described in Section 1.6.3 of this chapter. A problem intrinsic to nested simulation is that the number of simulation becomes enormous. There are many studies on algorithms for accelerating nested simulation. As examples, see Devineau and Loisel (2009) and Gordy and Juneja (2010), among others. 1.7

Validity of VaR

VaR is widely used as a risk measure in regulatory guidelines in the banking sectors. However it is known that VaR at a particular confidence level does not tell us the loss beyond that level, although the validity of VaR has been studied mathematically and financially in many papers. In this section, we make some brief remarks about the properties of VaR as a downside risk measure and define expected shortfall as a coherent risk measure. We address these issues basically according to the Bank for International Settlement (BIS, 2004b) for tail risk and according to Artzner et al. (1999) for coherent risk measures. For this purpose, we denote by X the value of a portfolio at time T , and by ρ a risk measure for X. Then, ρ(X) represents the size of the risk of X measured by ρ. Naturally, the portfolio is allowed to sell some options, which creates the possibility that X will take a negative value. To avoid confusion, we note that a large positive ρ(X) represents a larger risk than a small positive or negative ρ(X). We remark that up to the previous section, risk measures have been defined in terms of the profit–loss measures of a portfolio, such as sensitivity, convexity, and VaR. In this section, we work with a risk measure for X that does not represent profit or loss. In this context, we define a VaR of Vα at confidence level 1 − α over a holding period T as Vα (X) = − inf{x|P(X ≤ x) > α}.

(1.52)

Tail risk To explain the tail risk, we consider two portfolios, whose prices at time T are denoted by A for one portfolio and B for the other. We use A and B as the names of these portfolios. We assume that A and B have the same VaR at confidence level 1 − α, so that Vα (A) = Vα (B). In Fig. 1.9, the solid line shows

26 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

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A B

Downside risk



Value of portfolio

Figure 1.9: VaR and downside risk of two portfolios. Downside risk of B is larger than that of A.

the distribution of future value of A, and the dashed line shows that of B. We see that the downside risk of portfolio B is larger than that of A. This shows that V aRα does not capture the difference in downside risk between portfolios, as exemplified by A and B; this risk is noted in BIS (2004b) as tail risk. Under VaR regulations, risk managers will choose portfolio B in preference to A because there is a larger opportunity for profit in B. Thus, VaR might mislead risk managers who optimally control their portfolios. For further argument along these lines, see Basak and Shapiro (2001) and Yamai and Yoshiba (2002), among others. Coherent risk measure Furthermore, other conceptual problems with VaR are claimed by Artzner et al. (1999) and others. To address this, we define four properties according to Artzner et al. (1999) as follows. 1) Translation invariance: For a real number c, it holds that ρ(X + c) = ρ(X) − c. Namely, if we add cash in the amount c to the portfolio, then the risk is reduced by c. 2) Subadditivity: ρ(X1 + X2 ) = ρ(X1 ) + ρ(X2 ). This can be summarized

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as “a merger does not create extra risk.” 3) Positive homogeneity: For positive values of λ, it holds that ρ(λX) = λρ(X). 4) Monotonicity: If X1 ≤ X2 , then ρ(X1 ) ≥ ρ(X2 ). When a risk measure ρ satisfies these four properties, it is said to be coherent. Artzner et al. (1999) states that risk measures ought to be coherent for use in financial risk management. Example 1.7.1: Subadditivity Next, we present a famous example from Artzner et al. (1999), showing that percentile-based VaR does not satisfy subadditivity. Consider an investor who has two digital options a and b on a stock price with expiry T . The initial price of option a is u > 0, and the payoff at T is defined by the stock price ST at T and the strike price U as follows. • The investor pays 1000 if ST > U . • The investor pays nothing if ST ≤ U . This scenario is equivalent to selling a digital call option with strike price U . The initial price of option b is l > 0, and the payoff at T is defined by ST and the strike price L, with L < U , as follows. • The investor pays nothing if ST ≥ L. • The investor pays 1000 if ST < L. This is equivalent to selling a digital put option with strike price L. We assume that P(ST < L) = P(ST > U ) = 0.008.

(1.53)

Then, the VaR at the 99% confidence level for each position is given by V0.01 (a) = −u and V0.01 (b) = −l. Note that these negative values indicate a profit. Consider a portfolio a + b. Since P((ST < L) ∪ (ST > U )) = 0.016,

(1.54)

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ESα(B) A B



Value of Portfolio

ESα (A)

Figure 1.10: Expected shortfalls of two portfolios the VaR of a + b is V0.01 (a + b) = 1000 − u − l.

(1.55)

We see that V0.01 (a + b) > V0.01 (a) + V0.01 (b); that is, subadditivity is not satisfied. This is caused by the fact that both V0.01 (a) and V0.01 (b) fail to capture downside risk beyond the confidence level. Expected shortfall An expected shortfall (also called tail conditional expectation and conditional VaR) at a confidence level 1 − α over the holding time T is defined by ESα (X) = E[−X| − X ≥ Vα (X)].

(1.56)

Fig. 1.10 illustrates the value distributions of A and B, which are the as same as in Fig. 1.9. We additionally include ESα (A) and ESα (B) in the figure, which intuitively shows that ESα (B) is larger than ESα (A). This relation looks more reasonable for risk management than the percentile-based VaR does. We remark that the expected shortfall is not always a better risk measure for financial risk management than the percentile VaR is, which is addressed

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in Yamai and Yoshiba (2002, 2005), and so on. Dynamic risk measure The VaR and expected shortfall are static measures. However, in financial institutions, the risk profile changes with time. With this change, a dynamic risk measure has been studied successfully. This subject is beyond the scope of this book. For details, readers are recommended to consult Hardy and Wirch (2004) or Kriele and Wolf (2012). Remark Although the investigation into the best measure to use is of practical importance, it is undisputed that these risk measures should be calculated under the real-world measure. And so, the problem of risk measure should be studied in connection with the problem of real-world modeling. Hence, the study of real-world modeling will remain a central subject of risk management. Additionally, if future financial regulations require a risk measure other than VaR to be used, the theory in this book will still be applicable.

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Chapter 2 FUNDAMENTALS OF STOCHASTIC ANALYSIS

Abstract: This chapter briefly summarizes basic concepts of stochastic calculus, using intuitive examples. First, the fundamentals of probability spaces are introduced by working with a simple example of a stochastic process. Next, stochastic processes are introduced in connection with a natural filtration and a martingale. Then, we introduce a stochastic integral and Ito’s formula, which is an important tool for solving stochastic differential equations. Finally, we address some fundamental examples of stochastic differential equations, which simply model the price process of a financial asset. Although these subjects are applied in practice to interest rate modeling, the definitions are given for the one-dimensional case for the sake of simplicity. We complement this with some basic results for multi-dimensional cases in Section 2.7, at the end of this chapter.

Keywords: Abstract Bayes’ rule, Augmented filtration, Brownian motion, Conditional expectation, Distribution function, Euler approximation, Equivalent measure, Exponential martingale, Filtered probability space, Girsanov theorem, Ito’s formula, Market measure, Martingale, Measure change, Natural filtration, Probability space, Quadratic covariation, Random walk, Real-world measure, Risk-neutral measure, Radon–Nykodim derivative, Random variable, Sample space, Stochastic differential equation, Stochastic integral, Stochastic process.

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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This chapter briefly summarizes basic concepts of stochastic calculus, using intuitive examples. We remark that the descriptions in this section are not as rigorously given as they would be in academic textbooks in this field. For readers who are more interested in theoretical aspects, there are many excellent works in the field of probability. Chung and Williams (1990), Karatzas ¨ and Shreve (1998), Oksendal (2003), Pliska (1997), and Shreve (2004) are particularly recommended. In connection with interest rate modeling, the reader may find it useful to consult Baxter and Rennie (1996), Bj¨ork (2004), Cairns (2004), Gatarek et al. (2007), Hull (2000), James and Webber (2000), Shreve (2004), and Tavella (2003), among others. 2.1

Probability Space

Probability space We denote by Ω a set of all possible outcomes, called the sample space, and an element w ∈ Ω, called a sample. The family F is a collection of subsets of Ω and is assumed to be a σ-algebra. A stochastic event A is represented as an element in F. The function P is a probability function defined on F such that P(Ω) = 1 and P(φ) = 0; this is also called a probability measure. For A ∈ F, P(A) indicates probability of event A occurring. From these, we characterize a probability space by a triplet (Ω, F, P) . Example 2.1: Sample space In financial engineering, a sample w ∈ Ω is regarded as a history of up and down moves of a stock price observed in a fixed period. This makes Ω the set of up and down sequences of the stock price. If a period consists of two days {t1 , t2 }, then Ω will contain four samples and be given by Ω = {uu, ud, du, dd},

(2.1)

where we represent by u and d the up and down movements, respectively, of the stock price. Accordingly, F consists of all subsets of Ω, F = {φ, Ω, uu, ud, du, dd, {uu, ud}, {uu, dd, }, · · · },

(2.2)

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which means that F has 16 (= 24 ) events.

Real-world measure In financial market analysis, the probability measure derived from observation of market prices is called the market measure, with many synonyms, such as historical measure, physical measure, actual measure, and real-world measure. We regard historical data as being observed under the real-world measure. Recall the previous example. If our observation estimates that the probability of an up move is constant at 0.5 and the down-move probability is 0.5, then we have P(uu) = P(ud) = P(du) = P(dd) = 0.25 (2.3) Accordingly, for an arbitrary event A ∈ F, it is possible to give P(A). This probability measure P is an example of the real-world measure. Risk-neutral measure Another important measure is the risk-neutral measure, which is used in practice for derivatives pricing. Many financial engineers regard the risk-neutral measure as artificial or just a mathematical concept used in arbitrage pricing. In economics, the risk-neutral measure is explained as occurring in a virtual world where risk-neutral investors are living and trading. This world is sometimes called the “risk-neutral world.” To contrast with the risk-neutral world, the “world of the real-world measure” is called a “real world.” In financial institutions, many practitioners work in the risk-neutral world, and they may be unaware of the real-world measure, or may not need to know the difference between the two measures. This book focuses on interest rate modeling in the “real world.” It is natural to assume the uniqueness of the real-world measure for working with the financial market as a mathematical model. The risk-neutral measure is artificially implied from the real-world measure, and so several kinds of risk-neutral measures exist and are applied for various objectives. These include the spot measure, spot LIBOR measure, forward measure, and terminal measure; after Chapter 4 these will be used often.

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Equivalent measures Let P and Q be probability measures defined on F. The measure Q is said to be absolutely continuous with respect to P if Q(A) = 0 holds for any A ∈ Ω for which P(A) = 0. Additionally, P is said to be equivalent to Q if P and Q are mutually absolutely continuous. We sometimes denote by P∼Q

(2.4)

that P is equivalent to Q. In derivatives pricing, the risk-neutral measure is the most important example of a measure equivalent to the real-world measure. 2.2

Random Variables

Expectation and variance A random variable X is defined as a real-valued measurable function on a sample space Ω. We regard interest rates, bond prices, and stock prices on future days as random variables throughout this book. The expectation of a random variable X under measure P is defined as  E[X] = X(w)P(dw). (2.5) w∈Ω

The expectation is sometimes referred to as the mean. The variance of X is defined as  v(x) = X(w)2 P(dw). (2.6) w∈Ω

The variance V (X) is a measure of how widely X varies from E[X]. Expectation is linear on linear combinations of random variables. Let Y be another random variable. As an example, for constants a, b, and c, the expectation of the linear combination a + bX + cY is given by E[a + bX + cY ] = a + bE[X] + c[Y ].

(2.7)

The variance of a + bX + cY is given by V (a + bX + cY ) = b2 V (X) + c2 V (Y ) + 2bc Cov(X, Y ),

(2.8)

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Stochastic Analysis

where Cov is the covariance between X and Y and is defined as  (X(w) − E[X])(Y (w) − E[Y ])P (dw). Cov(X, Y ) =

(2.9)

w∈Ω

The correlation coefficient ρ(X, Y ) is defined as Cov(X, Y ) . ρ(X, Y ) =  V (X)V (Y )

(2.10)

For any constants a and b, it holds that ab Cov(X, Y ) Cov(aX, bY ) = ρ(aX, bY ) =  V (aX)V (bY ) a2 b2 V (X)V (Y ) = ρ(X, Y ).

(2.11)

Hence, the correlation coefficient is invariant to the magnitudes of a and b. The correlation coefficient can be used as a measure of the strength of the linear relationship between X and Y . Naturally, it holds that −1 ≤ ρ ≤ 1. Distribution function For a random variable X, we define a real-valued function F (x) on R by F (x) = P(X < x).

(2.12)

F is called the cumulative density function. We assume in this book that F (x) is always differentiable at x, and so the derivative with respect to x, dF (x) , (2.13) dx is well defined on R. The function f (x) is called the density function or distribution function of X. That is, the distribution function f (x) represents the probability that the random variable X takes the value x. The most important distribution function in financial risk management is the normal distribution; for the definition of that distribution, see Appendix A.1. As examples, the forward rates in Heath et al. (1992) and the short rate in Hull and White (1994) have normal distributions. These are introduced in Chapter 4. For a positive-valued random variable X, if log X has a normal distribution, then X is said to be lognormally distributed. For example, an asset price f (x) =

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Takashi Yasuoka

equation in Section 2.6 implies that the future asset price is lognormally distributed, which means that the price will take positive values. Similarly, the interest rate should be modeled so as to take positive interest. To give another example, the LIBOR market model, as introduced by Miltersen et al. (1997), Brace et al. (1997), Musiela and Rutkowski (1997), Jamshidian (1997), and others, assumes that the forward LIBOR has a lognormal distribution. This model is practically important as a positive interest rate model for derivatives pricing, which will be introduced in Chapter 5. The Radon–Nikodym theorem The relation between two measures P and Q can be represented by applying the Radon–Nikoym theorem, which asserts that there exists a measurable function z ≥ 0 almost everywhere (a.e.), such that for arbitrary A ∈ Ω it holds that   Q(dw) = z(w)P(dw). (2.14) A

A

Here, z(ω) is called a Radon–Nikodym derivative and denotes dQ . (2.15) dP To distinguish the measure used when calculating the expectation, we denote by E Q [ ] the expectation under Q. Then, for a random variable X, the expectation under Q is represented by   Q X(w)Q(dw) = X(w)z(w)P(dw) E [X] = z=

w∈Ω

= E[zX].

w∈Ω

(2.16)

In this book, the Radon–Nikodym theorem will be applied often to construct another measure Q from the original measure P by using a measurable function that has z(ω) > 0 a.e. 2.3

Stochastic Process

For arbitrage pricing, the dynamics of market price change is represented by a stochastic process. To introduce the concept of a stochastic process, we develop a probability space such that stochastic processes are well defined in the

Stochastic Analysis

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space, where the key is a filtration used to define a stochastic process. Filtration For a fixed time τ > 0, let Ft∈[0,τ ] denote an increasing subset of F; that is, Fs ⊂ Ft for 0 ≤ s < t ≤ τ . We assume that each Ft is a σ-algebra. Under this definition, Ft∈[0,τ ] is called an augmented filtration, or a filtration for short. Along these lines, (Ω, F, Ft∈[0,τ ] , P) is called a filtered probability space, or a probability space for short. Without loss of generality, we may set F = Fτ , which means that (Ω, F, Ft∈[0,τ ] , P) can be represented by a triplet (Ω, Ft∈[0,τ ] , P) for convenience. In particular, we use a filtration chosen such that it represents a flow of information; namely, Ft comprises the information about all events observed up to time t. An example of such a filtration is given in Example 2.3.1. Stochastic process Recall the probability space introduced in Example 2.1.1, that is, the up and down moves of stock price. Stock price movement between days is uncertain and unpredictable, making it an example of a discrete-time stochastic process. In a continuous-time setting, a stochastic process Xt is defined as a set of random variables indexed by time t, t ≥ 0 and defined on a probability space (Ω, Ft∈[0,τ ] , P). The process Xt is said to be continuous if almost all sample paths are continuous in t. Additionally, Xt is said to be adapted if Xt is Ft measurable at an arbitrary time t ∈ [0, τ ]. This is called Ft -adapted for short. In this book, we work mostly with adapted processes. Example 2.3.1: Adapted processes and natural filtration Regarding the up and down moves of a stock price in Example 2.1.1 as a stochastic process, we construct a filtration. At time 0, we set F0 = {φ, Ω}. This represents the absence of information about stock price moves at t = 0. The sets U and D are defined by U = {uu, ud},

D = {du, dd}.

(2.17)

Set U shows a stock price rise up at time 1, but the movement direction at time 2 is not known. Similarly, D indicates that the stock price falls at time

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1, but the movement direction at time 2 is not known. Then, the information that we have at time 1 is denoted by F1 , and we write F1 = F0 ∪ {U, D}.

(2.18)

Next, U U, U D, DU , and DD are each defined by a one-sample path such that U U = {uu}, U D = {ud}, DU = {du}, DD = {uu}.

(2.19)

Accordingly, F2 is defined as a σ-algebra such that F2 = F0 ∪ {U U, U D, DU, DD, U U ∪ U D, DU ∪ DD · · · }.

(2.20)

In this way, a filtration is constructed from Fi∈{0,1,2} . We see that Fi represents the information on the stock price up to time i for i = 0, 1, 2. In this sense, Ft is explained as the flow of information observed up to time t. The filtration Fi∈{0,1,2} is generated by the stock price process; such a filtration is called a natural filtration. Then this is the adapted process corresponding to the natural filtration. Random walks A random walk, in this context, is a process of up/down movements (jumps); a random walk is a well-known example of a stochastic discrete-time process. For example, the stochastic process in Example 2.3.1 is a random walk. Given an integer n, let us consider a random walk wn (t) such that • the initial value wn (0) = 0; • the value jumps at time i/n, i = 1, · · · ; • the probabilities of up and down jumps are always equal, at 0.5 each; and √ • the size of jumps is fixed at 1/ n. Fig. 2.1 exhibits examples of a random walk for n = 4 and n = 50. It is known that if we let n go to infinity, then wn (t) becomes a Brownian motion.

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Stochastic Analysis n=4

5 4

1

3

0 0

2

1

2

3

4

5

-1

1

-2

0 -1

n=50

2

0

1

2

3

4

5 -3

Figure 2.1: Random walks for n = 4 and n = 50 Because of this, we may view the case of n = 50 in Fig. 2.1 as approximately exhibiting a path of Brownian motion. Brownian motion The Wiener process is the most important example of a stochastic process in financial engineering and is also referred to as a Brownian motion. A stochastic process Wt , t ≥ 0 is called a Brownian motion when the following properties hold. • W0 = 0. • For 0 ≤ s < t, Wt and Wt − Ws are independent from each other. • Wt − Ws is normally distributed with mean 0 and variance t − s. In this book, we always assume that Ft∈[0,τ ] is a natural filtration generated by Brownian motion. 2.4

Martingales and Conditional Expectation

Conditional expectation For an Fτ -measurable random variable X and t ≤ τ , E[X|Ft ] is the expectation of X conditional on the information Ft . Sometimes, E[X|Ft ] is denoted by Et [X] for simplicity, and called conditional expectation for short. Intuitively,

40 Interest Rate Modeling for Risk Management

Takashi Yasuoka

Xt Xs = a

E[Xt|Xs = a]

X0

0

s

t

Time

(a) Conditional expectation E[ Xt ¦ Xs = a]

Xt E[Xt|A]

A∈Fs X0

0

s

t

(b) Conditional expectation E[ Xt ¦ A]

Figure 2.2: Two examples of conditional expectation

Time

Stochastic Analysis

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the conditional expectation Et [X] is the expectation of X implied from the information up to time t. Let us consider the meaning of conditional expectation using Fig. 2.2, where Fig. 2.2(a) exhibits three sample paths, all taking the value a at time s. Under this condition, the expectation of Xt at time t ≥ s is denoted by E[Xt |Xs = a],

(2.21)

which is shown in Fig. 2.2(a). The conditional expectation with respect to all values of Xs is defined by E[Xt |Xs ]. We can see that E[Xt |Xs ] is a random variable with respect to the state Xs at time s. Next, Fig. 2.2(b) exhibits three sample paths, all having the same path up to time s. According to Example 2.3.1, we may regard these paths as an event A ∈ Fs . The expectation of Xt under the condition A is denoted by E[Xt |A] ; A ∈ Fs ,

(2.22)

which is also written in Fig. 2.2(b). Then, E[Xt |Fs ] represents the expectation, conditional on all events of Fs . This is a random variable with respect to all event in A ∈ Fs . Fig. 2.2(b) suggests that the conditional expectation E[Xt |Fs ] ; 0 ≤ s ≤ t

(2.23)

represents the expectation of Xt implied from the information up to time s, as previously mentioned. Properties of conditional expectation Here, we introduce three basic properties of conditional expectation that we will use repeatedly later. 1) The following equation is known as the law of iterated expectations: Es [Et [X]] = Es [X] ; 0 < s < t < τ.

(2.24)

2) For an Ft -measurable random variable Y , the following relation is sometimes employed. Et [XY ] = Y Et [X].

(2.25)

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3) Note that Et [X] is a random variables on Ft for t > 0. In particular, E0 [X] is precisely the usual expectation of X, that is, E0 [X] = E[X]. In particular, both relations (2.24) and (2.25) hold under more general conditions. For details, see, for example, Jacod and Protter (2003) or Shreve (2004). Bayes’ rule Bayes’ rule is a well-known formula that can be used to calculate conditional probability. This is generalized to conditional expectation as follows, with the result called the abstract Bayes’ rule. For details, see, for example, Bj¨ork (2004), or Musiela and Rutkowski (2011). Proposition 2.4.1 For an arbitrary t, 0 < t < τ , let X be an Ft -measurable random variable. Set z = dQ/dP. Then, it holds that EtQ [X] =

Et [Xz] , Et [z]

(2.26)

where E Q [ ] denotes the expectation under Q. Martingale A stochastic process Xt is called a martingale if it holds for arbitrary t > s > 0 that Xs = Es [Xt ]. A martingale is a mathematical model of a fair game. Specifically, the averaged future value of Xt cannot be predicted at time s from the past trajectories. Next, we show two important examples of martingale processes. Example 2.4.1 (1) Let X be an Fτ -measurable random variable. From the law of iterative expectation, it holds that Es [Et [X]] = Es [X],

s ≤ t ≤ τ.

(2.27)

Then, regarding Et [X] as a stochastic process, Et [X] is a martingale. (2) The Brownian motion Wt is known as a martingale. Then, it holds for 0 < s < t that Es [Wt ] = Ws .

Stochastic Analysis

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Measure change and martingales Arbitrage pricing theory is constructed by using the properties of martingales, which allow us to sometimes change measure from the real-world measure to a risk-neutral measure, or vice versa. In such a measure change, the following proposition is repeatedly applied so as to preserve the martingale structure. Let Q be a measure equivalent to P. We define a stochastic process zt by   dQ z t = Et . (2.28) dP It follows that z0 = 1 and zt > 0 almost surely (a.s.). Substituting dQ/dP into X of (2.27), we have      dQ dQ = Es . (2.29) E s Et dP dP Substituting (2.28) into the above equation, it follows that Es [zt ] = zs . Thus, zt is a P-martingale. The following proposition holds. Proposition 2.4.2 Let Xt be a stochastic process. Then, Xt is a Q-martingale if and only if Xt zt is a P-martingale. Proof We may set zτ = dQ/dP. If Xt is a Q-martingale, then Xt = Et [Xs ] for 0 ≤ t < s ≤ τ . From (2.24), (2.25), (2.28), and the abstract Bayes’ rule, we have Et [Xs zτ ] Et [zτ ] Et [Es [Xs zτ ]] Et [Xs Es [zτ ]] = = Et [zτ ] Et [zτ ] Et [Xs zs ] = . zt

Xt = EtQ [Xs ] =

(2.30)

It holds that Xt zt = Et [Xs zs ], and so Xt zt is a P-martingale. The converse is shown in the same way as above. ✷

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2.5

Takashi Yasuoka

Stochastic Integral

This section summarizes basic definitions and results for stochastic differential equations but omits proofs. For details and a rigorous description see any of ¨ Chung and Williams (1990), Karatzas and Shreve (1998), Oksendal (2003), and Shreve (2004). Stochastic integral Let Wt be a Brownian motion under a measure P. For a stochastic process t Xt satisfying 0 |Xs |2 ds < ∞ a.s., the stochastic integral of Xt with respect to Wt is defined by  t n−1  Xs dWs = lim Xti {Wti+1 − Wti }, (2.31) Δ→0

0

i=1

where 0 = t0 < · · · < tn = t and Δt = max0≤i≤n−1 (ti+1 − ti ). The following properties are well known. 1) If Xt is a deterministic process, that is, generated by a deterministic function on t, then it holds that  t  E Xs dWs = 0. (2.32) 0

2)

t 0

Xs dWs has normal distribution with mean 0 and variance  t   t 2 |Xs |2 ds. E ( Xs dWs ) = 0

(2.33)

0

Generally, for a function σ(t, Xt ) with inputs t and Xt , and regarding σ(t, Xt ) as a stochastic process, the stochastic integral is defined by  t σ(s, Xs )dWs . (2.34) 0

Euler approximation In practice, it is typically necessary to compute the stochastic integral numerically. Euler approximation (or integration) is a widely used technique for

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Stochastic Analysis

this problem, and furthermore plays an important role in theoretical studies of numerical analysis. Indeed, we shall apply the Euler approximation for investigating a real-world model after Chapter 4. For a continuous stochastic process Xt , we denote the stochastic integral of t Xt with respect to W by 0 Xs dWs . By the definition of the integral, we have, from (2.31),  t n−1  Xs dWs = lim X(ti ){W (ti+1 ) − W (ti )}. (2.35) Δ→0

0

i=1

Then, it follows for a sufficiently short period Δt that  Δt Xs dWs ≈ X0 {WΔt − W0 } 0

= X0 WΔt √ = ΔtX0 W1 ,

(2.36)

where W1 is identified as a random variable having a standard normal distribution. Hence, by substituting numbers generated according to a standard normal distribution into W1 , we approximately obtain the distribution of  Δt Xs dWs . 0 Example 2.5: Martingale τ For future time τ , let ft be an Ft -adapted process satisfying E[ 0 fs2 ds] < ∞. t From this, it is known that Xt = 0 fs dWs is a martingale with respect to P. Since Xt = 0, it holds for t ≤ T that  t E[ fs dWs ] = 0. (2.37) 0

The difference of the above from (2.32) is that Xt in (2.32) is deterministic, but fs in the above is a stochastic process. 2.6

Stochastic Differential Equation

Let us consider a stochastic differential equation of the form dYt = μ(t, Xt )dt + σ(t, Xt )dWt

(2.38)

46 (FRQRPLFV&XUUHQWDQG)XWXUH'HYHORSPHQWV9RO QG(G

with initial value Y0 . The solution of this equation is given by  t  t Yt = Y0 + μ(s, Xs )ds + σ(s, Xs )dWs . 0

Takashi Yasuoka

(2.39)

0

A stochastic process of the form (2.39) is called an Ito process, where μ(t, Xt ) is called the drift term and σ(t, Xt ) is called the diffusion term. For convenience, we denote μ(t, Xt ) and σ(t, Xt ) by μ(t) and σ(t), respectively. Sometimes, we will denote them by μt and σt or by μ and σ when the meaning is unambiguous. In the same way, we may denote a stochastic process Xt by X(t) or X for simplicity and according to the situation. Example 2.6: Constant drift and diffusion For positive constants a and b, consider the following process: dYt = adt + bdWt .

(2.40)

This is the most trivial example of a stochastic differential equation. From (2.39), the solution is given by Yt = Y0 + at + bWt .

(2.41)

Here, Yt is a sum of the deterministic drift at and a Brownian motion bWt with the diffusion term b. Covariance process For stochastic processes X and Y , the quadratic covariation process is denoted by X, Y t . In particular, if X and Y are Ito processes such that dXt = μ1 (t)dt + σ1 (t)dWt

(2.42a)

dYt = μ2 (t)dt + σ2 (t)dWt ,

(2.42b)

then X, Y is obtained by  t σ1 (s)σ2 (s)ds. X, Y t =

(2.43)

0

In practice, we empirically estimate the covariance among several asset prices from historical observation. This covariance corresponds to a discretetime version of the quadratic covariation process.

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Stochastic Analysis

Ito’s formula Let Xt be an Ito process defined by dXt = μ(t)dt + σ(t)dWt . For a C 2 -function g(t, x), g(t, X) is also an Ito process and ∂g 1 ∂ 2g ∂g dt + dX + dXdX, ∂t ∂x 2 ∂x2 which is called the Ito formula. We have dg(t, Xt ) =

(2.44)

dXdX = {μ(t)dt + σ(t)dWt }2 .

(2.45)

Using the relations dtdt = dtdWt = dWt dt = 0

(2.46)

and dWt dWt = dt, we have dXdX = σ 2 (t)dt. Then, the Ito formula can be represented as ∂g ∂g σ 2 (t) ∂ 2 g + {μ(t)dt + σ(t)dWt } + dt ∂t ∂x 2 ∂x2   ∂g ∂g σ 2 (t) ∂ 2 g ∂g = + μ(t) + dt + σ(t) dWt . 2 ∂t ∂x 2 ∂x ∂x

dg(t, Xt ) =

(2.47)

It is a notable property of the Ito formula that the process g(t, Xt ) is also represented by another Ito process. Some stochastic differential equations can be solved by applying Ito’s formula as in the following. Asset price equation Let μ and σ be positive constants. Consider the following equation with initial value X0 dXt = μdt + σdWt Xt

(2.48a)

dXt = μXt dt + σXt dWt .

(2.48b)

or

To solve this equation, we set g(t, x) = log x, and then we have ∂g = 0, ∂t

∂g 1 = , ∂x x

1 ∂ 2g = − 2. 2 ∂x x

(2.49)

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Takashi Yasuoka

Substituting these into (2.47), and σXt and μXt into σ(t) and μ(t), respectively, in (2.47), we have dg(t, X) = d log Xt   μXt σ 2 Xt2 1 σXt = 0+ − dWt dt + 2 Xt 2 Xt Xt   σ2 dt + σdWt . = μ− 2

(2.50)

Hence, Xt log = X0



σ2 μ− 2



t + σWt .

Thus, the solution of (2.48a) is obtained as    σ2 μ− t + σWt . Xt = X0 exp 2

(2.51)

(2.52)

From (2.51), we see that Xt is lognormally distributed. A stochastic process represented as (2.52) is called a geometric Brownian motion. The dynamics of asset price is sometimes represented by the equation (2.48a). As an example, the Black–Scholes model represents a stock price process in this form. Next, we consider the more general case where μ(t) and σ(t) are deterministic functions of t. Then, a stochastic differential equation can be described as dXt = μ(t)Xt dt + σ(t)Xt dWt . By the same arguments as above, the solution is given by    t   t σ 2 (s) dt + Xt = X0 exp μ(s) − σ(s)Ws . 2 0 0

(2.53)

(2.54)

Furthermore, this result is generalized for the equation with stochastic coefficients such that dXt = μt Xt dt + σt Xt dWt ,

(2.55)

Stochastic Analysis

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where μt and σt are adapted processes. Similarly to (2.54), the solution of (2.55) is given by   t    t |σs |2 Xt = X0 exp ds + σs dWs . μs − (2.56) 2 0 0

These results are applied for interest rate modeling in the multi-dimensional case and for bond price process equations. We will present analogous results for the multi-dimensional case in Section 2.7 of this chapter. Exponential martingale Let σt be an adapted process and consider an Ito process such that dXt = σt Xt dWt . Substituting μt = 0 into (2.56), we have   t   t |σs |2 ds + σs dWs . Xt = X0 exp − 2 0 0

(2.57)

(2.58)

It is known that Xt is a martingale under P; this type is referred to as an exponential martingale. For details, see Shreve (2004). We will apply this property for arbitrage theory in Chapter 3, where a relative bond price process will be represented in the above form. The Girsanov theorem Recall that we are working in the probability space (Ω, Ft∈[0,τ ] , P), where Ft∈[0,τ ] is the natural filtration generated by the Brownian motion Wt . Let Yt be an Ito process of the form dYt = μ(t)dt + dWt . Let Q be another probability measure defined by the Radon–Nikodym derivative such that    τ  dQ 1 τ 2 = exp − μs dWs − μ(s) ds , (2.59) dP 2 0 o

where Q is equivalent to P. The Girsanov theorem states that Yt is a Brownian motion with respect to Q. Wt is a martingale with respect to P, and Yt is also a martingale with respect to Q. To distinguish the measure where a process is a martingale, we say that Wt is a P-martingale for short. From the same reason, a Brownian motion with respect to P is referred to as a P-Brownian motion for simplicity. The treatment of Q-martingale and Q-Brownian motion is analogous.

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2.7

Takashi Yasuoka

Multi-dimensional Stochastic Process

In the previous section, the definitions and results for the stochastic differential equation were given for the one-dimensional case. Naturally, most of these can be generalized to the multi-dimensional case, which is where we will work for arbitrage pricing in this book. We briefly introduce a multi-dimensional case of the Ito formula and the Girsanov theorem. Stochastic integral Let (Ω, Ft∈[0,τ ] , P) be a probability space. Let Wt = (Wt1 , · · · , Wtd )T and Xt = (Xt1 , · · · , Xtd )T be a d-dimensional Brownian motion and a d-dimensional stochastic process, respectively. Here the superscript T denotes the transpose. We denote by Xt dWt the standard Euclidean inner product of Xt and dWt in Rd . The stochastic integral of Xt with respect to W is defined by  t d  t  Xs dWs = (2.60) Xsi dWsi . 0

i=1

0

Ito’s formula Let Xt be an n-dimensional Ito process dXt = μt dt + σt dWt , where μt is an n-dimensional process and σt is a process with n × d matrixes as entries. Let g(t, x) = (g1 (t, x), · · · , gp (t, x))

(2.61)

be a set of C 2 -functions. With this, the Ito formula is described as dgk (t, Xt ) =

n n  1  ∂ 2 gk ∂gk ∂gk dt + dX i + dX i dX j , ∂t ∂x 2 ∂x ∂x i i j i=1 i,j=1

(2.62)

for k = 1, · · · , p. Example 2.7: Multi-dimensional Ito’s formula Setting p = 1 for the above, we consider an Ito process dYt = μt Yt dt + σt Yt dWt ,

(2.63)

Stochastic Analysis

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where σt is a d-dimensional process and Wt is a d-dimensional Brownian motion. We set g(t, x) = log x, and then it follows that ∂g ∂g 1 ∂ 2g 1 = 0, = , = − 2. 2 ∂t ∂x x ∂x x Substituting these into (2.62), we have dYt 1 dYt dYt − Yt 2Yt2 1 (μt Yt dt + σt Yt dWt )2 = μt dt + σt dWt − 2 2Yt |σt |2 = μt dt + σt dWt − dt 2   |σt |2 dt + σt dWt , = μt − 2

(2.64)

dg(t, Y ) = d log Yt = 0 +

where we use (2.46) and dWti dWtj = δij dt. Consequently, we have  t     t |σs |2 σs dWs . μs − ds + Yt = Y0 exp 2 0 0

(2.65)

(2.66)

The difference of (2.66) from (2.56) is that σs dWs is the standard Euclidean inner product in Rd . This result will be used frequently for multi-factor interest rate models later. The multi-dimensional Girsanov theorem The Girsanov theorem is also available for the measure change in a multidimensional case as follows. Let Yt be a d-dimensional Ito process dYt = μ(t)dt + dWt , where μt is a d-dimensional process and Wt is a d-dimensional Brownian motion. Let Q be a probability measure defined by    τ  τ 1 dQ μi (s)dWsi − μi (s)2 ds . (2.67a) = exp − dP 2 o 0 i i Then, Q is equivalent to P, and Yt is a Brownian motion under Q. For convenience, we represent the above in the following form:   τ   dQ 1 τ 2 = exp − μs dWs − |μ(s)| ds . dP 2 0 o

(2.67b)

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5

Chapter 3 ARBITRAGE THEORY

Abstract: This chapter summarizes arbitrage theory in the framework of martingale theory. First, we introduce an arbitrage-free market and arbitrage price for the general asset market, where the key concepts are the state price deflator and a martingale. Next, a num´eraire and a num´eraire measure are introduced to generalize arbitrage theory. Accordingly, we will see that the arbitrage price does not vary with the choice of num´eraire. Next, we work with a bond market where the bond prices are represented by Ito processes. For this, the market price of risk is introduced to ensure the arbitrage-free condition in the market. The market price of risk widely plays an important role in traditional interest-rate models, as an example, which will appear in the basic theory of the HJM model in Chapter 4. The estimation of the market price of risk is the most important subject of this book and is studied after Chapter 6.

Keywords: Accumulated contribution rate, Arbitrage opportunity, Arbitrage pricing, Bond market, Change of num´eraire, Complete market, Contribution rate, Eigenvalue, Equivalent martingale measure, Num´eraire, Num´eraire measure, Market price of risk, Option pricing, Pricing kernel, Principal component, Relative price process, Risk-neutral measure, Risk-neutral pricing, Riskneutral valuation, Self-financing trading strategy, State price deflator, Timehomogeneous short rate model.

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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Takashi Yasuoka

This chapter briefly summarizes arbitrage theory in the framework of martingale theory. The arbitrage pricing theory has been mathematically constructed in Harrison and Kreps (1979) and Harrison and Pliska (1981). For details, refer to those works, or to others such as Pliska (1997) and Munk (2011). For interest-rate models, refer to Bj¨ork (2004), Cairns (2004), or Munk (2011), among others. 3.1

Arbitrage Pricing

Asset market Let τ > 0 be a time horizon, and let (Ω, Ft∈[0,τ ] , P) be a probability space, where Ft∈[0,τ ] is the augmented filtration and P is a real-world measure. Let M = {Sit }i∈Λ denote a family of price processes of assets, where M may contain very wide classes of asset, for example, a stock market or a bonds market. We remark that each Sit is not assumed to be an Ito process. We denote by Λ a set of indices of assets. For example, a finite set of numbers {1, · · · , n} can be used for a finite asset market, and a time interval [0, τ ] can be used for the family of bonds with maturity T ≤ τ . In this context, M is referred to as a market. Self-financing trading strategy For simplicity, we work with a finite asset market M = {Sit }i=1,··· ,n . A trading strategy is an Rn -valued process θt = (θ1t , · · · , θnt ), where each θit represents a volume of asset i. Then, the value of the portfolio p(t) at t is given by p(t) = θt St =

n 

θit Sit .

(3.1)

i=1

θ is said to be a self-financing trading strategy if dp(t) = θt dSt .

(3.2)

In the self-financing trading portfolio, we neither add nor withdraw money. The next example shows the meaning of (3.2).

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Arbitrage Theory

Example 3.1 Let T be a fixed time in the future. Suppose a market that consists of only two assets: a risk-free asset and a risky asset. Let r be a risk-free rate for a period [0, T ]. We use S1t to denote an investment at this rate. Then, it holds that S1t = exp(rt). We use S2t to denote a price process of a risky asset with S20 = 1. We set a trading strategy as θ1t = −1,

θ2t = 1.

(3.3)

These indicate borrowing 1 at the rate r and buying the risky asset by paying 1, respectively. The initial value of this portfolio is equal to 0, and it holds that dp(t) = −dS1t + dS2t .

(3.4)

This is an example of the form (3.2), and so it is a self-financing trading strategy. Arbitrage opportunity An arbitrage opportunity is a self-financing trading strategy in which the value of the initial portfolio is not greater than 0 but the value at time T is nonnegative with probability 1 and positive with nonzero probability. In Example 3.1.1, the initial value of the portfolio is 0, where we neither pay nor borrow any money for the principal. If it holds that p(T ) ≥ 0 with probability 1, then we absolutely do not lose. Additionally, if p(T ) > 0 with positive probability, then we have a chance for profit with no risk of loss. Such cases are arbitrage opportunities. Arbitrage-free market The market M is said to be arbitrage-free if no arbitrage opportunities exist. That is, if starting from no money allows no profitable choices. Therefore, the arbitrage-free condition is deeply related to the risk-free investment. The arbitrage-free condition is mathematically represented as a stochastic process ξt > 0 (a.s.) with ξ0 = 1 such that the processes ξt Sit are P-martingales for all i ∈ Λ. Typically, ξ is called the state price deflator (or the pricing kernel).

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Takashi Yasuoka

We remark here that the state price deflator might not exist as more than a theoretical concept, and it is not obvious how to numerically calculate it. Practically, the state price deflator roughly represents a kind of discount factor, as explained below. Arbitrage pricing To introduce arbitrage pricing in an arbitrage-free market, we consider a security whose payoff at time T ≤ τ is given by an FT -measurable random variable X. We denote by pt the arbitrage price of this security at time t, 0 ≤ t ≤ T . Then, pt is given by pt =

1 Et [ξT X], ξt

(3.5)

where ξt is the state price deflator. The financial meaning of this price is that we can find no arbitrage opportunity in our market at this price. In particular, the time 0 price of this security can be represented as p0 = E[ξT X].

(3.6)

If we regard ξT as the discount factor to the time T , then (3.6) suggests that the arbitrage price is equal to the expectation of the present value of the payoff at T . Although this interpretation seems intuitively to be comprehensive, we have no feasible information for determining the value of ξT explicitly, and we do not have a method to calculate it at time 0. Thus, (3.5) is not sufficiently practical for use in pricing; in the next section, we develop a version of (3.5) that is more usable. 3.2

Change of Num´ eraire

This section presents the most basic argument in arbitrage theory. We show that the arbitrage price is not affected by choice of num´eraire, and make an arbitrage pricing system that is more practical. First, we present Theorem 3.2.1, which gives an arbitrage-free condition in a martingale approach. Theorem 3.2.1(2) will be applied to constructing the HJM model in Chapter 4. Additionally, Theorem 3.2.1(2) is generalized to Theorem 3.2.2, which will be

Arbitrage Theory

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used in constructing the LIBOR market model in Chapter 5. Num´ eraire and num´ eraire measure A num´eraire process is a price process of a traded asset that takes positive value almost surely. For example, a risk-free bond can be used as a num´eraire. Consider the market M = {Sit }i∈Λ , which appeared in the previous section. Let an arbitrary i ∈ Λ be fixed, and define zt by Sit z t = ξt . (3.7) Si0 From the assumption of an arbitrage-free market, zt is a P-martingale with z0 = 1 and zt > 0 a.s. Let Pi denote a measure defined by the Radon–Nikodym derivative such that   dPi Siτ = z τ = ξτ . (3.8) dP Si0 From (3.7), we have zt /Sit = ξt /Si0 . For any arbitrary j, multiplying both sides of the above by Sjt gives zt

ξt Sjt Sjt = . Sit Si0

(3.9)

The arbitrage-free assumption implies that the right side is a P-martingale. From Proposition 2.4.2, we see that Sjt /Sit is a Pi -martingale for all j. Obviously, Pi is equivalent to P. This is called the Si num´eraire measure. Accordingly, we have the following theorem. Theorem 3.2.1 (1) If M is arbitrage-free, and it holds that Si > 0 a.s. for some i ∈ Λ, then Sjt /Sit is a Pi -martingale for all j ∈ Λ. (2) If there exists Sit > 0 a.s. for some i ∈ Λ such that Sjt /Sit is a Pi martingale for all j ∈ Λ, then M is arbitrage-free. Proof Case (1) has already been shown. It is sufficient to prove only case (2). From (3.7) and (3.8), we see that   ξt Sit dPi zt = E t . (3.10) = dP Si0

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Takashi Yasuoka

Since Sjt /Sit is a Pi -martingale for every j, Proposition 2.4.2 implies that Sjt zt /Sit is a P-martingale. Accordingly, we have, from (3.10), that Sjt zt Sjt ξt Sit Sjt ξt = = . (3.11) Sit Sit Si0 Si0 Hence, ξt Sjt is a Pi -martingale for every j, and so M is arbitrage-free. This completes the proof. ✷ Sjt /Sit is referred to as a relative price process. The condition that the relative price process Sjt /Sit is a Pi -martingale means that investment in asset j is a fair game when comparing with investment in asset i in the “world of Pi .” In this context, the num´eraire is regarded as a kind of benchmark for investment in the world of Pi . Arbitrage pricing under Pi Recalling the security with payoff X at T ≤ τ , the price process of this security is given by (3.5). Taking Si as a num´eraire, we denote by E i [ ] the expectation under the Si num´eraire measure Pi . From (3.8), (3.10), and the abstract Bayes’ rule, we have       1 Xzτ Sit Xzτ X i = Et = Sit Et Sit Et SiT SiT Et [zτ ] SiT zt   Xzτ Si0 . (3.12) = Et SiT ξt From (3.7), it follows that        Xzτ X ξτ Siτ X ξτ Siτ Et = Et = Et ET SiT SiT Si0 S S     iT i0  ξT Sit X ξT SiT X ET = Et = Et SiT Si0 SiT Si0 Et [XξT ] = . Si0 Substituting (3.13) into (3.12), we have   X Et [ξT X] Et [XξT ] Si0 i Sit Et = = SiT Si0 ξt ξt = pt .

(3.13)

(3.14)

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Arbitrage Theory

Consequently, the time t price of this security satisfies   X i pt = Sit Et . SiT

(3.15)

If we take another Sk as the num´eraire, it also holds that   X k pt = Skt Et . (3.16) SkT Therefore, the arbitrage price is not affected by the choice of the asset used as num´eraire. The following theorem gives us a more general form of Theorem 3.2.1(2).

Theorem 3.2.2 Let Sit , i ∈ Λ be a price process of some asset in the market M with Sit > 0 (a.s.). If there exists a measure Q equivalent to P such that Sjt /Sit is a Q-martingale for all j ∈ Λ, then M is arbitrage-free.

Proof

We set dQ zτ = , dP

zt = Et



dQ dP



(3.17)

for 0 ≤ t ≤ τ . This means that zt is well defined and a P-martingale. We define ξt by Si0 (3.18) ξt = zt . Sit By an argument similar to the proof of Theorem 3.2.1, it follows that for 0≤s≤t≤T   zt Sjt Es [ξt Sjt ] = Si0 Es S   it Sjt = Si0 Es Et zτ S   it Sjt = Si0 EtQ (3.19) Es [zτ ] Sit for any j ∈ Λ. By assumption, Sjt /Sit is a Q-martingale. Then, (3.19) becomes   Sjs ξs Sis Q Sjt Si0 Es Es [zτ ] = Si0 Sit Sis Si0 = ξs Sjs . (3.20)

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Combining (3.19) with (3.20), we see that Es [ξt Sjt ] = ξs Sjs . That is, all ξt Sjt , j ∈ Λ are P-martingales. Thus, M is arbitrage-free. This completes the proof. ✷ This theorem shows that if there exists some num´eraire asset and a num´eraire measure equivalent to P such that all relative prices are martingales with respect to the num´eraire measure, then the market is arbitrage-free. Usually, Q is referred to as an equivalent martingale measure. The difference between Theorem 3.2.1(2) and Theorem 3.2.2 may seem slight, but it is significant in practice. Indeed, in order to define a new measure Pi in Theorem 3.2.1(2), we need to have the state price deflator. For Theorem 3.2.2, in contrast, we may define ξ artificially by (3.18), and are not concerned with whether ξ is the state price deflator. Succinctly, we need not determine the state price deflator to build an arbitrage-free model. Given this context, Theorem 3.2.2 is more practical than Theorem 3.2.1(2). Moreover, the arbitrage price is defined similarly to that under Pi as follows. Arbitrage pricing under Q Let us consider the arbitrage pricing under Q. Recall again the security whose payoff at T is given by X. Set zt and ξt as in (3.17) and (3.18), respectively. Since zt is a P-martingale, the same calculation as (3.12) implies that     XzT 1 X Q = Sit Et Sit Et SiT SiT Et [zT ]   XzT Si0 . (3.21) = Et SiT ξt Similarly to the implication of (3.13), we have   Et [XξT ] XzT = . (3.22) Et SiT Si0 Substituting this into (3.21), we have, from (3.5), that   Et [XξS ] X Q = = pt . Sit Et SiT ξt Hence, the price of this security is given by   X Q . pt = Sit Et SiT

(3.23)

(3.24)

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Arbitrage Theory

This form is an analogy to the price formula (3.15) derived under Pi , and (3.23) shows that the price is equal to that defined under the real-world measure. In contrast to the real-world measure, Pi and Q are referred to as risk-neutral measures. We remark that the price in the form (3.24) is given by a num´eraire and the num´eraire measure is given without use of the state price deflator. Therefore, the arbitrage pricing within Theorem 3.2.2 can be practically executed. This approach is referred to as a risk-neutral valuation or risk-neutral pricing. As a result, for arbitrage pricing, we may choose an arbitrary num´eraire according to the properties of the asset. This technique is referred to as change of num´eraire. Remark In this section, we assume no specific properties of an asset price process. In this context, the arbitrage pricing theory has been developed more generally in a mathematical sense. This book does not cover these subjects further. For more advanced study, see Musiela and Rutkowski (2011), or Pliska (1997), among others. 3.3 Market Price of Risk The previous section presented a general framework of arbitrage pricing without use of a Brownian motion. Next, we introduce another framework for a bond market where bond prices are represented by Ito processes. In this setting, we consider the market price of risk to define the arbitrage-free market. Let Wt be a d-dimensional P-Brownian motion defined on (Ω, Ft∈[0,τ ] , P), and let {Bit }i∈Λ be a family of price processes of zero-coupon bonds. We denote by B = {Bit }i∈Λ the bond market. In this section, we assume that values Bit are represented by Ito processes such that dBit = μit dt + υit dWt , i = 1, · · · , n, (3.25) Bi where μit is a one-dimensional adapted process, and υit is a d-dimensional adapted process satisfying  T  T |μit |ds < ∞, (3.26) |υit |2 ds < ∞, 0

0

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Takashi Yasuoka

respectively. We sometimes denote these as μi and υi , omitting the subscript t. From the Ito formula (2.66), we have Bit = Bi0 exp

 t  0

|υi |2 μi − 2



ds +



t

υi dWs 0



(3.27)

for each i. Also, ξt denotes an adapted process defined by dξt = −rt dt − ϕt dWt , ξt

(3.28)

with ξ0 = 1, where rt is a one-dimensional adapted process, and ϕt is a dT T dimensional adapted process such that 0 |r|ds < ∞ and 0 |ϕ|2 ds < ∞. We sometimes denote these as r and ϕ, omitting the subscript t. From (2.66), ξt is represented by  t     t |ϕ|2 −r − ds − ϕdWs . (3.29) ξt = exp 2 0 0 Arbitrage-free bond market In the bond market B, we have another criterion for the arbitrage-free condition. This is given in the following theorem. Theorem 3.3.1 Let Sit , i = 1, · · · , n and ξ be Ito processes defined by (3.25) and (3.28), respectively. Then, the market M is arbitrage-free if there exist r and ϕ such that μi − r = υ i ϕ

(3.30)

for every i. Proof

From (3.27) and (3.29), we have    t   t |ϕ|2 |υi |2 − ds + (υi − ϕ)dWs ξBit = Bi0 exp μi − r − 2 2 0 0   t    t |ϕ|2 |υi |2 υi ϕ − = Bi0 exp (υi − ϕ)dWs ds + − 2 2 0 0   t   t |υi − ϕ|2 (3.31) = Bi0 exp − ds + (υi − ϕ)dWs . 2 0 0

Arbitrage Theory

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The right-hand side is an exponential martingale, as mentioned in Section 2.6, and so ξBit is a martingale under P for all i ∈ Λ. Specifically, ξ is a state price deflator and B is arbitrage-free. This completes the proof. ✷ The arbitrage pricing in the market B is given in the same way as in Sections 3.1 and 3.2. In particular, ϕ is a market price of interest rate risk (hereinafter, market price of risk). If we obtain the market price of risk ϕ, then r is given from (3.30) by r = μi − υi ϕ. The state price deflator ξt is obtained by (3.28).

Complete market If the market price of risk exists and is unique, then the market is called complete. Originally, the complete market is defined in the sense of replication of options for option pricing and hedging. However, this book is interested in the uniqueness of the market price of risk, rather than in the replication problem. Therefore, in this book, a “complete market” means that the market price of risk is unique. Economic interpretation of the market price of risk To give an economic interpretation of the market price of risk, suppose that Wt is a one-dimensional Brownian motion. Then, the market price of risk is a scalar-valued process. From (3.30), we have μi − r ϕ= . (3.32) υi In traditional financial theory, r is regarded as a risk-free interest rate and μi is regarded as the expected rate of return for bond i. Then, μi − r indicates the excess return for bond i. Here, υi is a price volatility of bond i; that is, it is a kind of risk measure of bond i. The market price of risk is traditionally explained as the excess return per unit of risk for investment in bond i. The value r is referred to as the implied short rate. Furthermore, the existence of the market price of risk means that the excess return per unit of risk for each bond is uniform. That is, there is no possibility to find a profitable bond with respect to the risk–return trade-off. This is an intuitive explanation of Theorem 3.3.1. However the traditional interpretation is limited to a one-dimensional model. In contrast to this, Chapters 6 and 9 will theoretically present a perspective

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for interpretation of a multi-dimensional model.

Takashi Yasuoka

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Chapter 4 HEATH–JARROW–MORTON MODEL

Abstract: For management of interest rate risk, we create an interest rate scenario by using an arbitrage-free model of the bond market, which describes the evolution of the forward rate. With this understanding, this chapter addresses the forward rate model introduced by Heath et al. (1992) (hereinafter, HJM). Additionally, we introduce the short rate model introduced by Hull and White (1990), which we treat as a special case in the HJM model. In interest rate models, the option price is typically valuated under the riskneutral measure, and so these models have been developed as models specified under the risk-neutral measure. On the one hand, when we apply a model to risk management, we must use a model specified under the real-world measure. We consider this further by valuating the VaR of a simple example. On the other hand, to construct an interest rate model under the real-world measure, it is necessary to estimate the market price of risk. We briefly summarize some approaches to estimation of that price in the short rate models.

Keywords: Arbitrage-free, Contribution rate, d-factor model, Drift coefficient, Forward rate, HJM model, Hull–White model, Kalman filter, Market price of risk, Mean reversion rate, Principal component analysis (PCA), Realworld measure, Real-world model, Risk-neutral measure, Risk neutral valuation, Savings account, Short rate, Spot measure, State price deflator, Timehomogeneous, Volatility, Volatility component .

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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4.1

Takashi Yasuoka

Heath-Jarrow-Morton Framework

Interest rate model We introduced a bond market B in Chapter 3 that did not admit a term structure of interest rates. If, instead, we assume a term structure in the bond market, it becomes possible to relate the bond price to the interest rate. We represent the dynamics of bond price by a stochastic process and use this to specify the corresponding interest rates. Such a system is referred to as a term structure model of interest rates (or put simply, an interest rate model). A model specified by the dynamics of a short rate is referred to as a short rate model. A model specified by the dynamics of forward rates is referred to as a forward rate model. For management of interest rate risk, it is better to suppose various types of changes in the yield curve, and specifically to suppose changes in the forward rates. Because of this, the forward rate model is more useful in risk management than the short rate model is. This section briefly introduces the HJM model, which is the most general forward rate model. For additional details, readers are recommended to consult Cairns (2004), Munk (2011), or Shreve (2004), among others. For details on calibration, readers are recommended to consult Wu (2009). Forward rate process Let (Ω, Ft∈[0,τ ] , P) be a filtered probability space, where Ft∈[0,τ ] is the augmented filtration and P denotes the real-world measure. The instantaneous forward rate with maturity T observed at time t is denoted by f (t, T ). When the usage is unambiguous, f (t, T ) will be called the forward rate. Typically f (0, T ) represents an initial forward rate. We assume that the dynamics of f (t, T ) on (Ω, Ft∈[0,τ ] , P) is represented by df (t, T ) = α(t, T )dt + σ(t, T )dWt ,

(4.1)

where Wt = (Wt1 , · · · , Wtd )T is a d-dimensional P-Brownian motion, and α(t, T ) and σ(t, T ) are predictable processes satisfying some technical conditions. Here, σ(t, T ) = (σ 1 (t, T ), · · · , σ d (t, T ))T is a d-dimensional process. The second term, σ(t, T )dWt denotes the inner product of σ(t, T ) and dWt in

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Heath-Jarrow-Morton Model

Rd , specifically σ(t, T )dWt =

d 

σ l (t, T )dWtl .

(4.2)

l=1

Adopting this convention for simplicity, ab denotes the inner product of vectors a = (a1 , · · · , ad )T and b = (b1 , · · · , bd )T throughout. The process α(t, T ) is referred to as a drift coefficient. Typically, when α(t, T ) is positive (resp., negative), the forward rate rises (resp., falls) on average. The process σ(t, T ) is referred to as a volatility; it represents the magnitude of the forward rate diffusion. From these, the equation (4.1) represents a stochastic model in the dynamics of forward rates. In particular, if the model uses a d-dimensional Brownian motion as the above, then it is referred to as a d-factor model for discrimination of the dimension size. Instantaneous short rate and savings account From equation (4.1), it follows that  t  t f (t, T ) = f (0, T ) + α(s, T )ds + σ(s, T )dWs . 0

(4.3)

0

The instantaneous short rate (hereinafter, short rate) r is given by r(t) = f (t, t). Substituting t into T in equation (4.3), we have  t  t α(s, t)ds + σ(s, t)dWs . (4.4) r(t) = f (0, t) + 0

0

A savings account (or money market account) Bt is defined by  t  r(s)ds . Bt = exp

(4.5)

0

This represents continuous reinvestment at the short rate r(t). In other words, the savings account serves as an indicator of performance by the safest investment. In the HJM model, we take the savings account as a num´eraire. For example, if the short rate is assumed to be constant (i.e., r(t) = r), then we have Bt = exp{rt}. This is well known as the performance of the investment on the constant short rate r for the period [0, t]. This form was

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used as the example of a risk-free asset in Example 3.1. Bond price We denote by B(t, T ) the price at time t of a zero-coupon bond with maturity T, 0 ≤ T ≤ τ . From equation (1.14), it follows that   T  B(t, T ) = exp − f (t, u)du . (4.6) t

A bond market B is defined as a set of all bonds with maturity T, 0 ≤ T ≤ τ , where each price of a T -maturity bond is given by equation (4.6). If f (t, T ) is continuous in T , then B(t, T ) is differentiable with respect to T . Thus, it follows from equation (4.6) that

∂ log B(t, T ) = f (t, T ). (4.7) ∂T It is shown by Heath et al. (1992) that B(t, T ) can be represented as   t |υ(s, T )|2 ds log B(t, T ) = log B(0, T ) + r(s) + b(s, T ) − 2 0  t υ(s, T )dWs , (4.8) + −

0

where b(s, T ) and υ(s, T ) denote processes given by  T |υ(s, T )|2 α(s, u)du + b(s, T ) = − 2 s  T σ(s, u)du, υ(s, T ) = −

(4.9) (4.10)

s

respectively. Relative bond price Taking the savings account as a num´eraire, from equations (4.5) and (4.8), the relative bond price is represented by   t B(t, T ) |υ(s, T )|2 log = log B(0, T ) + ds b(s, T ) − Bt 2 0  t υ(s, T )dWs . (4.11) + 0

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Heath-Jarrow-Morton Model

We assume the existence of some d-dimensional process ϕt that satisfies b(t, T ) = υ(t, T )ϕt

(4.12)

for all T . Here, ϕt is called the market price of risk. From the above relation, we note that ϕt is independent of T . Substituting b(t, T ) into equation (4.11), it follows that  t   |υ(s, T )|2 B(t, T ) = B(0, T ) exp ds υ(s, T )ϕs − Bt 2 0   t (4.13) + υ(s, T )dWs . 0

t We set a process Zt as Zt = 0 ϕs ds + Wt . From the Girsanov theorem, there exists a measure Q, equivalent to P, such that Zt is a Brownian motion under Q. Specifically, equation (4.13) becomes    t  t B(t, T ) |υ(s, T )|2 = B(0, T ) exp − ds + υ(s, T )(ϕs ds + dWs ) Bt 2 0 0    t  t |υ(s, T )|2 = B(0, T ) exp − υ(s, T )dZs . (4.14) ds + 2 0 0 Since the right-hand side is an exponential martingale, the relative bond price B(t, T )/Bt is a martingale for all maturities T . It follows from Theorem 3.2.1(2) that the bond market is arbitrage-free. From these, we have obtained an arbitrage-free interest rate model; this is reflected by the following theorem. Theorem 4.1.1 If there exists a market price of risk ϕt satisfying equation (4.12), then the bond market B is arbitrage-free. In the framework of HJM, the savings account is taken as a num´eraire, and the measure Q is the equivalent martingale measure with respect to the savings account. Q is also referred to as a risk-neutral measure. Remark We remark that the sign in the definition of ϕt given in equation (4.12) is opposite that used in Heath et al. (1992); however, our usage follows the conventional formulation of the market price of risk, as equation (3.30). This will be verified in the form (4.22).

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4.2

Takashi Yasuoka

Arbitrage Pricing and Market Price of Risk

This section briefly studies some fundamental subjects in the HJM model, specifically, forward rate process, arbitrage pricing, the market price of risk, and state price deflator. Forward rate process Here let us represent the forward rate process under the risk-neutral measure Q. Differentiating equation (4.9) with respect to T , we have  T ∂b(s, T ) −α(s, T ) − σ(s, T ) σ(s, u)du = . (4.15) ∂T s From equations (4.10) and (4.12), it follows that −α(s, T ) − σ(s, T )



T

∂υ(s, T ) ϕt ∂T = −σ(s, T )ϕt .

σ(s, u)du = s

(4.16)

Substituting the above into equation (4.1), we obtain df (t, T ) = {−σ(t, T )υ(t, T ) + σ(t, T )ϕ(t)} dt + σ(t, T )dWt . Recall the Q-Brownian motion Zt = the above, we have

t 0

(4.17)

ϕs ds + Wt . Substituting this into

df (t, T ) = −σ(t, T )υ(t, T )dt + σ(t, T )dZt ,

(4.18)

where the drift under Q is completely determined by the volatility σ(t, T ). This form is the well-known forward rate process in the HJM model. For pricing interest rate derivatives, the dynamics of the forward rates are typically simulated by equation (4.18), and the bond pricing is performed by using this form. This method is essentially the same as used with short rate models. Arbitrage pricing in the HJM model According to the argument in Section 3.2, let us verify the arbitrage pricing within the HJM framework. Consider a security whose payoff at T ≤ τ is

Heath-Jarrow-Morton Model

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defined by a random variable X. From equation (3.24), the time t price of this security is given by   X Q pt = B t E t . (4.19) BT

Here, Bt denotes the time t price of the savings account. This formula has two salient points. • The savings account taken as the num´eraire is familiar as a risk-free asset for derivative traders. • It is not necessary to estimate the market price of risk. In particular, let us consider a case in which the short rate r(t) is assumed to be a constant process, as r(t) = r. In that case, it holds that B0 = 1 and BT = exprT . From these, the price formula (4.19) becomes p0 = exp{−rT }E Q [X].

(4.20)

This expression indicates that the price of this security is the present value of the expectation of the value X at time (T ) under the risk-neutral measure Q. This constant short rate assumption is widely used in the primary model for option pricing. For example, it is assumed for the Black–Scholes model of a stock option and the Black model of a cap (or floor). Because of this, the formulation (4.20) is known as the risk-neutral valuation. Market price of risk Theorem 4.1.1 assures the existence of an arbitrage-free market under the assumption about the market price of risk ϕt that b(t, T ) = υ(t, T )ϕt .

(4.21)

This form looks to be different from the original definition given in equation (3.30) as μit − rt = υit ϕ˜t .

(4.22)

However, the two are actually the same, as follows. Here, we denote by ϕ˜t the original definition to distinguish it from ϕ as defined in equation (4.21). Let us examine the property of ϕ in connection with equation (4.22).

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Applying Ito’s formula to equation (4.8), we have dB(t, T ) = (r(t) + b(t, T ))dt + υ(t, T )dWt . B(t, T )

(4.23)

From the implications of equation (3.25), the drift μ(t, T ) of the T -maturity bond is represented by μ(t, T ) = r(t) + b(t, T ). With this, b(t, T ) indicates the excess rate of return for investment in the T -maturity bond relative to the short rate r(t). Combining equation (4.21) with μ(t, T ), we have μ(t, T ) − r(t) = υ(t, T )ϕt .

(4.24)

This expression shows the same form as equation (4.22), and so ϕt has the same function as ϕ˜t (cf. the remark in Section 4.1). Negative value of the market price of risk The market price of risk typically takes a negative value, and we offer a conventional explanation for that. For example, we can find a description in Cairns (2004, p. 62). Consider a one-dimensional case of equation (4.24); specifically, consider the market price of risk ϕt as being expressed by ϕt =

μ(t, T ) − r(t) . υ(t, T )

(4.25)

Since μ(t, T ) represents the expected rate of return for the T -maturity bond, the difference μ(t, T ) − r(t) indicates the excess return rate for the investment in T -maturity bond relative to the short rate r(t). Mostly, it might hold that μ(t, T ) > r(t). Then ϕt will have the same sign as the value υ(t, T ). Since υ(t, T ) must be negative from its definition in equation (4.10), we may view the market price of risk as taking a negative value. However, the truth of this observation is somewhat questionable. Indeed, it is neither trivial to estimate the drift coefficient μ(t, T ) of the bond from a historical price data, nor is it certain that μ(t, T ) > r(t) can be proved. On this topic, we will develop a theoretical study of the value in the market price of risk for the multi-dimensional case in Chapters 6 and 7.

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State price deflator In the previous section, we obtained an arbitrage-free bond market B, where the relative bond prices are Q-martingale. Yet, we have not obtained the state price deflator. Let us derive that from the market price of risk and the short rate. We suppose that we have already obtained the market price of risk ϕt . We define an Ito process ξt by dξt = −r(t)dt − ϕt dWt , (4.26) ξt where r(t) is the short rate process as r(t) = f (t, t). From Ito’s formula, ξt is represented by    t   t |ϕs |2 (4.27) ϕs dWs . ds − −r(s) − ξt = exp 2 0 0 Also, the bond price B(t, T ) is obtained from equation (4.23) as   t  |υ(s, T )|2 ds B(t, T ) = B(0, T ) exp r(s) + b(s, T ) − 2 0   t + υ(s, T )dWs .

(4.28)

0

From these, ξt B(t, T ) is represented by   t  |υ(s, T )|2 |ϕs |2 + b(s, T ) − ds − ξt B(t, T ) = B(0, T ) exp 2 2 0   t + {−ϕs + υ(s, T )}dWs . (4.29) 0

Substituting equation (4.12) into the above, we obtain   t  |υ(s, T )|2 |ϕs |2 + υ(s, T )ϕs − ds ξt B(t, T ) = B(0, T ) exp − 2 2 0   t + {−ϕs + υ(s, T )}dWs . (4.30) 0

By the same calculation as in the proof of Theorem 3.3.1, we have  t |ϕs − υ(s, T )|2 ds ξt B(t, T ) = B(0, T ) exp − 2 0   t + {−ϕs + υ(s, T )}dWs . 0

(4.31)

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Takashi Yasuoka

This means that ξt B(t, T ) is an exponential martingale, and is thus a Pmartingale for all T . From this, ξt is the state price deflator. As a result, once we obtain the market price of risk ϕt , the state price deflator is given by equation (4.26), where we assign the short rate r(t) as the implied short rate. Real-world model and risk-neutral model When an interest rate process and bond price process are represented under P, as in equation (4.8), the corresponding market is referred to as a real-world model. When they are represented under Q, as in equation (4.14), the market is referred to as a risk-neutral model. Note that the risk-neutral model is not unique because the risk-neutral measure is determined according to the choice of num´eraire. In contrast, the real-world model is uniquely determined for a complete market. Null assumption on the market price of risk When we assume a null market price of risk, ϕ = 0, it holds that Zt = Wt from t the relation Zt = 0 ϕs ds + Wt . From this, we may see that Q = P, almost everywhere. Therefore making a null assumption about the market price of risk is equivalent to using a risk-neutral model. Applying this, we can find many papers that intend to valuate the interest rate risk by using an interest rate model. Many such papers begin with a theoretical setting by using a real-world model. However, at the final step of computing the risk, they assume a null market price of risk. This approach is equivalent to working with the risk-neutral model. One consequence of this is that many of us may be aware of neither the market price of risk nor the real-world measure in the business of derivatives pricing. Naturally in these circumstances, estimating the market price of risk has not been focused on for several decades. 4.3

Volatility and Principal Components

Approaches to volatility estimation This section introduces a method for constructing the volatility in the HJM

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Heath-Jarrow-Morton Model 0.06 0.05 0.04

Oct-10 Dec-10 Feb-11 Apr-

0.03 0.02 0.01 0 Spot

Time (years)

1

2

3

4

5

6

7

8

9

Figure 4.1: Implied forward rate curve in the U.S. Treasury market at October 2010, December 2010, February 2011, and April 2011. Yield data are the same as those for Fig. 1.3.

model. There are two major approaches to do so. One is a market approach; the other is a historical approach. In the market approach, the volatility is estimated such that the model implies option prices consistent with their market prices. In the historical approach, the volatility is constructed to represent a historical dynamics of an interest rate, for example, the short rate or the forward rate. Experimentally, these two approaches result in quite different volatility structures. When we calibrate the model for derivatives pricing, the market approach should be employed. In this, it is understood that historical volatility cannot explain market prices because the option prices are determined mostly by traders’ forecasts for the future market rather than by historical volatility. Therefore, adopting a historical approach will result in a model that misprices major derivatives. Such a model is not valid for derivatives trading. However, when we intend to calibrate a model for interest-risk-management, the historical approach is recommended, rather than the market approach. In the historical approach, principal component analysis (PCA) is a standard technique for reducing the dimensionality of the model. PCA will be repeatedly used in this book, and we introduce the construction of volatility by applying PCA. The fundamentals of PCA and the relevant linear algebra are

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given in Appendix B. Historical observation To address the historical approach, let us recall Fig. 1.3 in Section 1.3, which exhibited a historical chart of the implied forward rates in the U.S. Treasury market. Additionally, Fig. 4.1 exhibits some cross sections of those rates. Specifically, it shows the implied forward rate curves observed at October 2010, December 2010, February 2011, and April 2011 for the same U.S. Treasury yields as shown in Fig. 1.3. These graphs suggest that changes in forward rates f ( ·, Ti ), i = 1, · · · , n are closely correlated with each other. Succinctly, f ( ·, Ti ) moves almost uniformly in i, i = 1, · · · , n. As an example, consider a case in which the forward rate curve is always moved as a parallel shift, and the dynamics of the forward rate is described by a one-factor model as df (t, Ti ) = · · · dt + σdWt ; i = 1, · · · , n

(4.32)

where σ is a scalar constant and Wt is a one-dimensional Brownian motion. Here, we focus our argument on the volatility structure, which means that we may not exactly specify the second term. From Fig. 4.1, we may adopt the view that the one-factor model can simply approximate the change in the forward rates. Covariance matrix Let xi , i = 1, · · · , n with xi < xi+1 be a sequence of time length. At time t, the variable Ti = xi + t represents a maturity date of a bond i. Let Δt be a time interval to observe the forward rates. In the HJM model, the change in the forward rate f (t, Ti ) is specified as Δf (t, Ti ) = f (t + Δt, Ti ) − f (t, Ti ) with fixed maturity Ti . Using the time length xi , this is represented as Δf (t, t + xi ) = f (t + Δt, t + xi − Δt) − f (t, t + xi ).

(4.33)

We typically observe the change in the forward rate in this form with a fixed time length xi . Applying this form, the covariance matrix V = {Vij } is given by the observation of Δf ( , xi ) as Vij =

1 Cov(Δf ( , xi ), Δf ( , xj )). Δt

(4.34)

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Principal volatility component PCA is a statistical technique for reducing the dimensionality of the model. For the covariance matrix V , we have from equation (B.11) in Appendix B that ⎛ ⎞ (ρ1 )2 0 0 ⎜ ⎟ 1 n T .. Vij = (e1 , · · · , en ) ⎝ 0 (4.35) . 0 ⎠ (e , · · · , e ) , 2 0 0 (ρn ) where el = (el1 , · · · , eln )T is the lth principal component and (ρl )2 is the lth eigenvalue. In this, we assume that the eigenvalues are chosen such that ρ1 > ρ2 > · · · > ρn > 0, and |el |2 = 1 for all l. The first principal component explains the largest portion of the covariance V , and the second one explains the largest portion of the remaining covariance, and so on. The set of principal components is orthonormal from equation (B.4); in other words, the components are uncorrelated with each other (cf. Appendix B.1). Let σ l = (σ1l , · · · , σnl )T denote the lth principal volatility component (briefly, the lth volatility component) defined by σil = ρl eli . It follows from the orthonormality condition that l 2

|σ |

=

n 

(ρl eli )2

i=1

= (ρl )2 .

n  = (ρl ) (eli )2 2

i=1

(4.36)

This means that the lth eigenvalue (ρl )2 indicates the magnitude of the lth volatility component. Principal component in the forward rates Here we summarize the basic property of the principal components in connection with the changes of the forward rates. Using this, Litterman and Scheinkman (1991) applied PCA to U.S. Treasury yields. They showed that the first factor essentially represents a parallel change in yields; this is called a level factor. The second factor is called steepness (or slope), and the third factor is called curvature (or hump-shapedness). Empirically, these three factors explain most of the changes in the yields. This feature is generally observed in the bond market and widely applied for interest-rate-risk management.

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Additionally, a property similar to the above is observed in the forward rate dynamics. Specifically, the first principal component represents a parallel change in the forward rate curve change, the second component indicates a steepening or flattening, and the third component represents a curvature change. Fig. 4.2 illustrates a representative example of the first three components in the forward rate dynamics. In this figure, the bold line e1 shows the first principal component, which typically takes a strictly positive value. This component represents parallel change in the forward rate curve. The thin line e2 shows the second component, which takes positive values in short forward rates and negative values in long forward rates. This indicates the steepness of change. The dashed line e3 indicates the third component, which takes positive values in short and long forward rates and takes negatives value in middle forward rates. This represents curvature change. From these observations, the principal components are used to construct a scenario of the forward rate changes for use in risk management. A numerical example of the principal components will be shown in Chapter 10. e1

Term to maturity

e3 e2

Figure 4.2: Representative curves of the first three principal components of forward rate dynamics. The curve ei represents the ith principal component for i = 1, 2, 3.

Accumulated contribution rate When a d-factor model is constructed by using the first d principal components, we call it the first d-factor model. Throughout this book, we always work with the first d-factor model, which we abbreviate to d-factor model.

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A contribution rate of the lth principal component is defined by (ρl )2 . (4.37) (ρ1 )2 + · · · + (ρn )2 This indicates the portion of the lth principal component in the covariance. An accumulated contribution rate Cd for the first d principal components is defined by Cd =

(ρ1 )2 + · · · + (ρd )2 . (ρ1 )2 + · · · + (ρn )2

Dimensionality reduction When we work with an n-factor model, the by an n × n matrix chosen such that ⎛ ⎞ ⎛ ⎛ ⎞ df (t, T1 ) ··· σ11 ⎜ ⎟ ⎜ .. ⎟ ⎜ . .. ⎝ ⎠ = ⎝ . ⎠ dt + ⎝ .. . σn1 ··· df (t, Tn )

(4.38)

volatility structure is represented ⎞⎛ ⎞ dWt1 . . . σ1n .. .. ⎟ ⎜ .. ⎟ , . . ⎠⎝ . ⎠ . . . σnn dWtn

(4.39)

where σil = ρl eli for i, l = 1, · · · , n. The accumulated contribution rate is a traditional criterion used when reducing the dimensionality of the forward rate model. When Cd is almost equal to 1 for an integer d > 0, then the above equation is successfully approximated by the d-factor model. It is empirically known that the first three-factor model adequately approximates the n-factor model of equation (4.39). When an observation such as this is known to hold, we may work with a three-factor model. The equation (4.39) is remarkably reduced to a three-factor model as ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ df (t, T1 ) ··· σ11 σ12 σ13 dWt1 ⎜ ⎟ ⎜ .. ⎟ ⎜ . .. .. .. ⎟ ⎝ dW 2 ⎠ . (4.40) ⎝ ⎠ = ⎝ . ⎠ dt + ⎝ .. . . . ⎠ t 3 1 2 3 dW df (t, Tn ) σn σ n σn ··· t Thus, it remains to determine the drift term; this will be studied in Chapter 6. 4.4

The Hull–White Model

In the early days, many stochastic models were introduced to describe the dynamics of the short rate. As examples, see Cox et al. (1985; hereinafter,

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CIR), Ho and Lee (1986), Hull and White (1990), and Vasicek (1977), among others. A strong point of these models is their parsimoniousness. Additionally, these models are described by affine term structures. For details of affine models, readers are recommended to consult Duffie and Kan (1996), Bj¨ork (2004), or Munk (2011). It is known that the Ho–Lee model and the Hull–White model are special cases of the Gaussian HJM model. The Hull–White model, in particular, is one of the most popular models in many financial institutions. Following along these lines, this section introduces the Hull–White model as a special case of the HJM model. Short rate process Let us consider a one-dimensional process of the short rate r(t) represented by  σ  dr(t) = κ θ(t) − r(t) + ϕt dt + σdWt , (4.41) κ where Wt is a one-dimensional Brownian motion under the real-world measure P; κ and σ are positive constants; θ(t) is a positive process; and ϕt denotes the market price of risk. It is empirically observed that the volatility of long-term interest rates is less than that of short term rates, reflecting a general phenomenon referred to as mean reversion. To model this feature, the rate at which r(t) reverts to θ(t) is the speed κ, called the mean reversion rate. t The savings account Bt = exp{ 0 r(s)ds} is taken as a num´eraire. We t set Zt = 0 ϕs ds + Wt . By the Girsanov theorem, there exists a risk-neutral measure Q equivalent to P such that Zt is a Brownian motion under Q. From these, the short rate r(t) is represented under Q as dr(t) = κ(θ(t) − r(t))dt + σdZt .

(4.42)

It is known that the price of a zero-coupon bond with maturity T is given by B(t, T ) = exp{−a(t, T ) − b(T − t)r(t)},

(4.43)

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where a(t, T ) and b(s) are defined by a(t, T ) = κ



T t

θ(s)b(T − s)ds +

σ2 b(T − t)2 4κ

σ2 + 2 {b(T − t) − (T − t)}. 2κ 1 b(s) = [1 − exp{−κ(s)}] κ For an arbitrary initial forward rate f (0, T ), we define θ(t) by θ(t) = f (0, t) +

σ2 1 ∂f (0, t) + 2 (1 − e−2κt ). κ ∂t 2κ

(4.44) (4.45)

(4.46)

With this, it is known that B(0, T ) coincides with the zero-coupon bond price implied from the initial rate f (0, T ). Substituting equation (4.46) into equation (4.42), r(t) is described as  σ2 1 ∂f (0, t) −2κt + 2 (1 − e ) − r(t) dt + σdZt . (4.47) dr(t) = κ f (0, t) + κ ∂t 2κ This is the formulation of the short rate process r(t) used in the Hull–White model. 

In this model, when θ(t) is determined by equation (4.46), B(0, T ) is equal to the initial bond price implied from arbitrarily given the forward rate f (0, T ). In this context, the Hull–White model is referred to as an arbitrage-free model. Among examples in Gaussian short rate models, the Ho–Lee model is known as an arbitrage-free model but the Vasicek model is not. For more about this, refer to the works of Ho and Lee (1986) and Hull and White (1990), or see Munk (2011) or Wu (2009), among others. Hull–White model as an HJM model The previous argument is derived from the short rate process (4.41). Next, we obtain the Hull–White model as a special case of HJM’s forward rate process. Consider a one-factor volatility in the HJM model given by σ(t, T ) = σ exp{−κ(T − t)},

(4.48)

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where σ and κ are positive constants. This volatility admits an exponentially decaying structure with respect to the time length T − t from the time t to the maturity T . From equation (4.18), the drift term is represented under Q by  T −κ(T −t) σ(t, T )υ(t, T ) = −σe σe−κ(T −s) ds t

2

σ −κ(T −t) e [1 − e−κ(T −t) ]. κ Substituting this into equation (4.18), f (t, T ) is described as  t 2 σ −κ(T −s) [1 − e−κ(T −s) ]ds e f (t, T ) = f (0, T ) + 0 κ  t +σ e−κ(T −s) dZs . = −

(4.49)

(4.50)

0

Since r(t) = f (t, t), putting T = t in the above results in expressing the short rate process as  t 2 σ −κ(t−s) r(t) = f (0, t) + e [1 − e−κ(t−s) ]ds κ 0  t + σe−κ(t−s) dZs 0  t σ2 −κt 2 −κt eκs dZs . (4.51) = f (0, t) + 2 (1 − e ) + σe 2κ 0 t Next, we define a process Xt by Xt = 0 eκs dZs , and a process g(t, Xt ) by g(t, Xt ) = f (0, t) +

σ2 (1 − e−κt )2 + σe−κt Xt . 2κ2

(4.52)

From equation (4.51), we see that g(t, Xt ) = r(t). In order to obtain dr = dg(t, Xt ) by Ito’s formula, we calculate the following: ∂f (0, t) σ 2 ∂g = + (1 − e−κt )e−κt − κσe−κt Xt ∂t ∂t κ ∂g = σe−κt ∂X ∂ 2g = 0. ∂X 2

(4.53a) (4.53b) (4.53c)

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From Ito’s formula, it follows that dr(t) = dg(t, Xt )   ∂f (0, t) σ 2 −κt −κt −κt = + (1 − e )e − κσe Xt dt + eκt σe−κt dZs ∂t κ    t ∂f (0, t) σ 2 −κt −κt −κt κs + (1 − e )e − κσe e dZs dt = ∂t κ 0 (4.54) +σdZs . Substituting equation (4.51) into the above, we have  ∂f (0, t) σ 2 −κt −κt dr(t) = + (1 − e )e dt ∂t κ   σ2 −κt 2 +κ f (0, t) + 2 (1 − e ) − r(t) dt + σdZs 2κ   ∂f (0, t) σ 2 −2κt + (1 − e ) − κr(t) dt = κf (0, t) + ∂t 2κ +σdZs . 

(4.55)

Setting θ(t) as θ(t) = f (0, t) +

σ2 1 ∂f (0, t) + 2 (1 − e−2κt ), κ ∂t 2κ

(4.56)

we have dr(t) = κ(θ(t) − r(t))dt + σdZs .

(4.57)

This coincides precisely with equation (4.46). From these, the Hull–White model with mean reversion rate κ and volatility σ is exactly an HJM model with volatility σ(t, T ) = σ exp{−κ(T − t)}. Distribution of short rate From equation (4.51), the short rate r(t) is normally distributed with mean σ2 E [r(t)] = f (0, t) + 2 (1 − e−κt )2 2κ Q

(4.58)

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under Q. The variance of the distribution is obtained from equation (2.33) as    t −κt κs 2 V ar[r(t)] = E (σe e dZs ) 0  t 2 e2κ(t−s) ds = σ 0

2

σ (1 − e−2κt ). (4.59) 2κ We note, in particular, that the variance is independent on the measure. We will address the calibration of this model for risk management in Chapter 8. =

4.5

VaR Computed in the Real-world

This section studies the reason that the VaR should be computed using a real-world model. For this purpose, the valuation of the VaR depends on the choice of measure. We use the following simple example to illustrate this. For simplicity, we assume a null discount rate in the following argument (i.e. the forward price is equal to the present price). Suppose a binary bond with expiry at time T and with payoff X at T is given as follows. ⎧ X=0 ⎨ If L > 5% at T, then (4.60) ⎩ If L ≤ 5% at T, then X = 1.01,

where L indicates the 6-month LIBOR at T . Succinctly, the payoff is determined by the level of the 6-month LIBOR at the expiry date. The price of this security is computed by using some interest rate model under some risk-neutral measure Q. For the model, we assume the probability distribution of L as ⎧ ⎨ Q(L > 5%) = 0.09% (4.61) ⎩ Q(L ≤ 5%) = 99.01%. With this distribution, the arbitrage price of this bond at t = 0 is calculated by (1.01 × 0.9901 + 0 × 0.0009) × 1 = 1.00

(4.62)

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because of the assumption of a null discount rate. We buy this bond at price 1.00. Let us valuate the 99% VaR of this bond for holding period T . We can sell this for the price 1.01 at time T at a probability of more than 99%. The 99% VaR is valuated as the profit of -0.01(=1.00-1.01) under Q. Next, we assume that historical observation estimates for the 6-month LIBOR are ⎧ ⎨ P(L > 5%) = 2% (4.63) ⎩ P(L ≤ 5%) = 98%,

where P denotes the real-world probability. From this, the 99% VaR is the loss of 1.00 (= 1.00 − 0). So the valuation of the VaR is absolutely affected by the measure. Even assuming a non-zero interest rate as the discount rate might affect little in this example. Table 4.1 summarizes these figures. Here, the second column exhibits the payoff of this security, and the third column exhibits the profit and loss. The last row exhibits the 99% VaR under the measures P and Q, respectively. Moreover, let us depart from the assumption of a null discount rate, and suppose an alternative num´eraire measure. This might imply yet another 99% VaR depending on the measure. These observations means that the VaR depends on the choice of the measure, in contrast to the arbitrage price, which does not vary with the choice of measure. Naturally, we have no probability more reliable than the real-world probability. Because of this, when we valuate the VaR by using some existing system at our institution, we must confirm whether the VaR has been calculated under the real-world measure and examine what method was employed to estimate the market price of risk. Taken all together, when we intend to construct a real-world model, it is possible to determine the volatility σ from the market data. It remains to estimate the market price of risk. This subject will be theoretically investigated in Chapters 6 and 8.

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Table 4.1: 99% VaR of the binary bond for holding period T. The initially invested amount is 1.00.

6-month LIBOR L > 0.05 L ≤ 0.05 VaR

4.6

Probability Payoff P/L Real-world Risk-neutral 0 -1.00 0.02 0.0009 1.01 0.01 0.98 0.9901 1.00 -0.01 (profit)

Estimation of the Market Price of Risk

In empirical analysis concerning the term structure of interest rates, we are observing historical data under the real-world measure. To give an example, when we use the Hull–White model, the dynamics of the short rate is described from equation (4.41) as  σ  (4.64) dr(t) = κ θ(t) − r(t) + ϕt dt + σdWt . κ

To calibrate this model such that this equation explains the historical dynamics of the short rate, we must estimate the parameters σ and κ and the market price of risk ϕt . In this way, we inevitably estimate ϕt as part of fitting any interest rate model with the historical dynamics of the interest rates. Along these lines, there are many studies on estimating the market price of risk in the field of economics. Some papers in this vein are Ahn and Gao (1999), Cheridito et al. (2007), Cochrane and Piazzesi (2010), De Jong (2000), and Duffee (2002). However, there are few papers that explicitly describe the method used in estimating the market price of risk in short rate models. It is even more difficult to find such papers that work with forward rate models. In this section, we briefly describe three approaches to estimating the market price of risk in short rate models. For a more advanced treatment of this subject, we study theoretical methods for estimating the market price of risk in the forward rate model from Chapter 6.

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Time-homogeneous short rate model To define the term “time-homogeneous” in relation to short rate models, let us consider the Vasicek model, which represents the short rate dynamics as drt = κ(θ − rt )dt + σdZt . (4.65)

Here, κ, θ, and σ are constants1 , and Zt is a one-dimensional Brownian motion under a risk-neutral measure. When κ, θ, and σ are assumed to be constant, as here, the model is called time-homogeneous For example, the CIR model and the so-called CKLS model (Chan et al., 1992) are other time-homogeneous models. Stanton (1997) introduced a numerical technique for studying the term structure of interest rates in time-homogeneous short rate models. Parameters, such as those above, are estimated empirically, and an approximation formula for the market price of risk is obtained. This approach is restricted to time-homogeneous models. As an example, the short rate dynamics in the Hull–White model was described by equation (4.64), above. Since the parameter θ(t) is not a constant in that model, the model is not time-homogeneous. However, time-homogeneous models are not arbitrage-free in the sense that it is impossible to fit a term structure with arbitrary initial forward rates in the manner described in Section 4.4. Specifically, this approach is difficult to apply to risk management. Multi-factor time-homogeneous model In the multi-dimensional case, Dempster et al. (2010) studied methods to estimate the market price of risk in a Gaussian three-factor model under the additional assumption that the model is a time-homogeneous Markov model. The three factors are defined as a long rate X1 , the slope of the yield curve X2 , and the short rate X3 . Their model is formulated as dXit = (μi − λi Xit + ϕi σi )dt + σi dWit , ; i = 1, 2

(4.66a)

dX3t = (κ(X1t + X2t − X3t ) + ϕ3 σ3 )dt + σ3 dW3t ,

(4.66b)

where ϕi denotes the market price of Xi -risk for i = 1, 2, 3. 1 The difference between the Vasicek model and the Hull–White model is that θ is assumed to be constant in the former.

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By the Euler approximation, the above equations are discretized as √ Xi(t+Δt) = Xit + (μi − λi Xit + ϕi σi )Δt + σi ΔtWi1 ; i = 1, 2 (4.67a) √ X3(t+Δt) = X3t + (κ(X1t + X2t − X3t ) + ϕ3 σ3 )Δt + σ3 ΔtW31 , (4.67b) where each of Wi1 (i = 1, 2, 3) is regarded as the standard normal distribution2 . The market price of Xi -risk ϕi is estimated by applying regression analysis to historical data. Dempster et al.’s approach is basically similar to the method introduced in Chapter 6 of this book. Broadly speaking, the regression formulation can be used to immediately and comprehensively estimate the market price of risk. However, the last equation in the above set contains the state variables X1t and X2t , and so it is not trivial to estimate ϕ3 in the manner described in Appendix C. To overcome this, the computationally intensive Kalman filter technique is applied for solving the regression model, and so this approach requires software to perform the calculation. Although this model is not arbitrage-free in the sense described above, a naive method of approximately fitting arbitrary initial forward rates is offered by three-factor models. Drift difference In an another approach from the above studies, Cheridito et al. (2007) estimates the market price of risk in affine models, with some implications consistent with the results of Chapters 6 and 9 of this book. We sketch their approach by using the Hull–White model as follows. The short rate r(t) is represented by dr(t) = aσκ (t, r)dt + σdZt ,

(4.68)

where aσκ (t, r) indicates the drift term in equation (4.47), and Zt is a Brownian motion under a risk-neutral measure Q. For this argument, the term aσκ (t, r) is referred to as a risk-neutral drift. 2

Wt denotes a Brownian motion with respect to time t. W1 has the standard normal distribution for t = 1. For convenience, we regard W1 as the standard normal distribution throughout this book. Then, Monte Carlo simulation is executed by substituting numbers generated according to a standard normal distribution into W1 .

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Under the real-world measure P, we may represent r(t) by dr(t) = b(t)dt + σdWt ,

(4.69)

where Wt is the Brownian motion under P, and b(t) is the drift estimated by the observation under P, which is referred to as historical drift here (in contrast to the risk-neutral drift). t We assert that Zt = 0 φs ds + Wt , where φt denotes the market price of risk. From this, we have that b(t) = aσκ (t, r) + σφt . Then, φt is obtained by φt =

b(t) − aσκ (t, r) , σ

(4.70)

where b(t) − aσκ (t, r) represents the difference between the historical drift and the risk-neutral drift. Equation (4.70) means that φt is characterized by the ratio given by the difference between historical drift and risk-neutral drift over short rate volatility. Specifically, if the historical drift is larger (resp., smaller) than the risk-neutral drift, then the market price of risk takes a positive (resp., negative) value. From this, the formulation as equation (4.70) has features in common with the results in Chapters 6 and 9, where we will give a theoretical interpretation of implications of the market price of risk.

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1

Chapter 5 LIBOR MARKET MODEL

Abstract: This chapter introduces the LIBOR market model, which is the standard model for derivatives pricing. Because the topic of this book is risk management, we do not deal with the details of pricing. Instead, this chapter introduces the model, focusing on the implications of the real-world model. First, we give a definition of the LIBOR market model, following Jamshidian (1997). Next, we define the LIBOR market model under the real-world measure (hereinafter, LMRW), and show, following the method of Yasuoka (2013b), that the model exists. Additionally we find the models under the spot LIBOR measure and under a forward measure that are implied by the LMRW. Finally, we verify the numerical differences of the LIBOR process according to choice of measure. The study on the real-world model will be developed in Chapter 9.

Keywords: Arbitrage-free, Arbitrage pricing, BGM model, Change of num´eraire, Deterministic volatility, Forward LIBOR, Forward measure, HJM model, LIBOR market model, LIBOR volatility: LMRW, Lognormal distribution, Market price of risk, Martingale approach, Positive interest rate, Real-world measure, Risk-neutral model, Spot LIBOR measure, State price deflator, Terminal measure .

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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5.1

Takashi Yasuoka

LIBOR Market Model

LIBOR market models The LIBOR market model was introduced by Miltersen et al. (1997), Brace et al. (1997; hereinafter, BGM), Musiela and Rutkowski (1997), and Jamshidian (1997). The notable points of this model are listed here: • The model has positive LIBOR.

• The model admits an arbitrary deterministic volatility structure. • The price formulae of a caplet and a floorlet are derived so as to be consistent with the corresponding Black’s price. • An approximated price formula for a swaption is derived. From these, the LIBOR market model has a usability advantage in calibration, and so it is widely applied as a standard model for derivatives pricing. As a particular example, the BGM model is the most well-known type of LIBOR market model, and is built in the HJM framework. The BGM approach requires a kind of differentiability for LIBOR volatility. It is impossible to satisfy this smoothness in practice because the volatility cannot be constructed except as a piecewise continuous, but not necessarily smooth, function. Because of this, the BGM model is not strictly supported in the HJM framework. For more advanced study of this problem, see Yasuoka (2001, 2013b). At one end of the spectrum of models, the approaches by Musiela and Rutkowski (1997) and Jamshidian (1997) stand on a martingale pricing theory, with no theoretical imperfections. However, their models are constructed under a risk-neutral measure without referring to the real-world measure. In this section and the next, we introduce the LIBOR market model as described by Jamshidian (1997). Because the topic of this book is risk management, pricing of derivatives is not addressed here at length. For a more advanced treatment of pricing, readers are recommended to consult Brigo and Mercurio (2007) or Gatarek et al. (2007). Similarly to the argument for the HJM model, when the LIBOR and bond prices are represented under a risk-neutral measure, we call the resulting system a risk-neutral model. When, instead, they are represented under P, the

Libor Market Model

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resulting system is referred to as a real-world model. Strict definitions of these terms will be given later. Definition of the LIBOR market model Let τ > 0 be a time horizon, and let (Ω, Ft∈[0,τ ] , P) be a probability space, where Ft∈[0,τ ] is the augmented filtration and P is the real-world measure. In this, Wt denotes a d-dimensional P-Brownian motion defined on (Ω, Ft∈[0,τ ] , P). Let 0 = T0 < T1 < · · · < Tn = τ be a sequence of times. Let Bi (t) denote a price process of a zero-coupon bond with maturity Ti such that Bi (Ti ) = 1 for i = 1, · · · , n, and let B = {Bi (t)}i=1,··· ,n denote the bond market. We remark that the market B is not assumed to be complete in this setup, again following Jamshidian (1997). We set δi as δi = Ti+1 − Ti ; i = 0, · · · , n − 1,

(5.1)

and denote by Li (t) the forward LIBOR observed at t over the period [Ti , Ti+1 ], which is given by 1 + δi Li (t) =

Bi (t) ; i = 0, · · · , n − 1. Bi+1 (t)

(5.2)

We first construct a positive LIBOR process under some measure Q equivalent to P. We assume that Li is represented by an Ito process such that dLi (t) = · · · dt + λi (t)dZt , Li (t)

(5.3)

where Zt is a d-dimensional Brownian motion under Q, and λi (t) is an Rd valued volatility of Li (t). The mathematical definition of the LIBOR market model (“the LIBOR model”) is given as follows. Definition 5.1 (Jamshidian model) B is called the LIBOR market model if the LIBOR process Li (t) is specified by a deterministic volatility λi (t), i = 1, · · · , n, such that Li (t) > 0, Bi (t) > 0 for i = 1, · · · , n, and B is arbitragefree.

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In this definition, the bond price process Bi (t) does not have to be an Ito process. Then, this model does not deal with the market price of risk and is constructed within the martingale framework introduced in Chapter 3. 5.2

Existence of LIBOR Market Model

The existence of the LIBOR model is shown in the following theorem. Theorem 5.2.1 For arbitrary deterministic volatility λi (t), i = 1, · · · , n − 1, the LIBOR market model exists. The LIBOR model can be constructed under any of several risk-neutral measures. Applying this, we will show the existence of the LIBOR model under the real-world measure in the next section, and show how the models are implied under other measures in Sections 5.4 and 5.5 of this chapter. It is thought that this approach is the simplest method of constructing the LIBOR market model for practical use. Therefore, we here only sketch Jamshidian’s LIBOR market model under a forward measure, omitting the proof. Let each of λi (t) be an arbitrary deterministic function in t for i = 1, · · · , n− 1. Consider the following equation: n−1  dLi (t) δj Lj (t)λi (t)λj (t) = dt + λi (t)dZt . Li (t) 1 + δj Lj j=i+1

(5.4)

Here, Zt is a d-dimensional Brownian motion with respect to a measure Q(∼ P). With this setup, the following proposition is given in Jamshidian (1997, Corollary 2.1). Proposition 5.2.2 The equation (5.4) admits a unique positive solution for an arbitrary initial condition Li (0) > 0 for all i. Further, Yi (t) = (1 + δi Li (t)) · · · (1 + δi Ln−1 (t)) is a Q-martingale. Let Bn (t) be an arbitrary bond price process such that Bn (Tn ) = 1 and 1 j=i (1 + δj Lj (Tj ))

Bn (Ti ) = n−1

(5.5)

Libor Market Model

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at each Ti . Accordingly, we define Bi (t) for i < n by n−1  Bi (t) = (1 + δj Lj (t)). Bn (t) j=i

(5.6)

From these, we see that Bi (Ti ) = 1 and the relation (5.2) is satisfied for all i.  By Proposition 5.2.2, n−1 j=i (1 + δj Lj (t)) is a Q-martingale for every i. Hence Bi (t)/Bn (t) is a Q-martingale for all i. Along these lines, Q is a Bn num´eraire measure and is referred to as a forward measure. As a result, the bond market B is arbitrage-free from Theorem 3.2.2. In this way, when the market B is represented under the forward measure Q, the market B is referred to as a forward measure model for simplicity. By changing the num´eraire to a different bond Bi , we can derive the Bi num´eraire measure in the manner described in Section 3.2. With this measure, we can obtain another forward measure model with respect to the Bi num´eraire measure. Non-uniqueness of the LIBOR model In the above sketch, the bond price process is allowed to have arbitrary value during the interval between each pair of successive maturity dates ending at Ti , because Bn (t) must satisfy the relation (5.5) only at each Ti . Thus, there exist infinitely many bond markets consistent with the LIBOR process (5.4). For some such markets, the bond price processes may be represented by Ito processes; for others, they may not. From this, the class of LIBOR market models is more relaxed than the class of HJM models. Comparison with HJM models In order to study the LMRW, it is worthwhile to verify the differences between LIBOR models and HJM models from the viewpoint of arbitrage theory. In an HJM model, the price processes of bonds are assumed to be Ito processes, and so the market price of risk is a critical factor in ensuring an arbitrage-free market. Additionally, the HJM framework for models is first constructed under the real-world measure P. Such a model is then reduced

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m(t) 6 5 4 3 2 1 δ0 0

δ1 T1

δ2 T2

t

δ3 T3

T4

T5

T6

Figure 5.1: Graph of m(t) to a model under a risk-neutral measure Q by applying the Girsanov theorem with the market price of risk. Therefore, the relation between P and Q is theoretically clear except for the estimation of the market price of risk. In contrast, the LIBOR market model is constructed under a risk-neutral measure in the martingale framework, as described above. Indeed, the price processes of bonds are not assumed to be Ito processes, and the market price of risk does not appear in the formulation. Instead, the model is initially constructed under the forward measure Q, where the relation between Q and P remains to be elucidated. We will address this relation in Section 5.5. 5.3

LIBOR Market Model under a Real-world Measure

Within the same setting as in Sections 5.1 and 5.2, we give a definition of the LMRW and show the existence of the model, following Yasuoka (2013a). Definition 5.3 The bond market B is called the LMRW when the following conditions are satisfied. 1. The LIBOR processes Li , i = 1, · · · , n, with Li (t) > 0, are represented under the real-world measure P such that each volatility λi (t) and the market price of risk ϕt are deterministic in t.

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Libor Market Model

2. The bond market B is arbitrage-free; here this means that Bi (t) ∈ B, i = 1, · · · , n and the state price deflator ξt are positive Ito processes represented under P. For this, we define a left-continuous function m(t) by m(t) = j, while t ∈ (Tj−1 , Tj ]. Succinctly, m(t) represents the index of the next maturity date1 . Examination of Fig. 5.1 may help to see the features of m(t). To show the existence of the LMRW, it is sufficient to give the simplest example for arbitrarily given volatility λ and market price of risk ϕ. For this, we define a process μ ¯(t) by μ ¯(t) = μ ¯(Tm(t) ) such that μ ¯(Ti ) =

1 δi−1

log{1 + δi−1 Li−1 (Ti−1 )}

(5.7)

¯(t) represents the yield for the shortest maturity at each time Ti . Specifically, μ bond, with the next maturity Tm(t) . As a consequence, μ ¯(t) is constant on each period (Ti−1 , Ti ], i = 1, · · · , n. Let ϕt be an arbitrarily given market price of risk such that ϕl is an Rd valued deterministic function with  τ |ϕt |2 ds < ∞. (5.8) 0

Let λi (t), i = 1, · · · , n be deterministic volatilities. We set χi (t) as χi (t) =

λi (t)δi Li (t) ; i = 1, · · · , n. 1 + δi Li (t)

Consider the following equation with the initial LIBOR Li (0) > 0, ⎧ ⎫ i ⎨ ⎬  dLi (t) = λi (t) χj (t) + λi (t)ϕt dt + λi (t)dWt , ⎩ ⎭ Li (t)

(5.9)

(5.10)

j=m(t)

for i = 1, · · · , n. It is known that the solution Li (t) exists uniquely and Li (t) > 0. We assume that bond price processes Bi (t), i = 1, · · · , n are Ito processes with initial values B0 (0) = 1 and Bi (0) =

i−1 

(1 + δj Lj (0))−1

(5.11)

j=0

1

The same function is used in BGM (1997, p. 130) and in Jamshidian (1997, p. 315).

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such that ⎧ ⎨

dBi (t) ¯(t) − = μ ⎩ Bi (t)

i−1 

χj (t)ϕt

⎫ ⎬ ⎭

j=m(t)

dt −

i−1 

χj (t)dWt .

(5.12)

j=m(t)

Under this setup, we give the following theorem, which shows the existence of the LMRW. Theorem 5.3.1 For arbitrary deterministic volatilities λi (t), i = 1, · · · , n and the deterministic market price of risk ϕt satisfying the condition (5.8), we define Bi (t), i = 1, · · · , n by equations (5.11) and (5.12). Then, the market B = {Bi (t)}i=1,··· ,n is the LMRW. Proof Since Li (t) > 0, μ ¯(t) is positive. Hence, μ ¯(t) is well defined. The proof is divided into the proofs of the following four claims. Claim 1: For all i, 1 + δi Li (t) =

Bi (t) Bi+1 (t)

a.s.

(5.13)

Proof of Claim 1 We may set B0 (0) = 1. From equation (5.11), we see that 1 + δi Li (0) =

Bi (0) ; i = 0, · · · , n − 1. Bi+1 (0)

(5.14)

Equation (5.12) admits a unique positive solution, as ⎧ ⎛ i−1 ⎨ t  ⎝ χ j ϕs − Bi (t) = Bi (0) exp μ ¯− ⎩ 0 j=m(s) ⎫  t  i−1 ⎬ χj dWs . − ⎭ 0 j=m(s)

2 ⎞     i−1   1  ⎠ ds χ j  2   j=m(s)

(5.15)

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It follows that  ⎧ ⎛ 2   i  t ⎨    Bi (t) 1 Bi (0)  ⎝χi ϕs +  = exp χ j  ⎩ 0 Bi+1 (t) Bi+1 (0) 2   j=m(s) 2 ⎞ ⎫    i−1  t ⎬    1  ⎠ −  χj  ds + χi dWt ⎭ 2 0  j=m(s) ⎧ ⎛ ⎞  t i ⎨ 2  Bi (0) |χi | ⎠ ⎝χi ϕs + χi = χj − exp ds ⎩ 0 Bi+1 (0) 2 j=m(s)   t χi dWs . +

(5.16)

0

Multiplying both sides of equation (5.10) by δi Li (t)/(1 + δi Li (t)), we have ⎧⎛ ⎫ ⎞ i ⎨ ⎬  dδi Li δi L i ⎝λ i = χj + λi ϕt ⎠ dt + λi dWt . ⎭ 1 + δi Li 1 + δi Li ⎩

(5.17)

j=m(t)

Substituting equation (5.9) into the right-hand side of the above, we obtain ⎛

dδi Li = ⎝ χi 1 + δi Li

i 

j=m(t)



χj + χi ϕt ⎠ dt + χi dWt .

The solution of this equation is uniquely given by ⎧ ⎛ i ⎨ t  ⎝ χj 1 + δi Li (t) = (1 + δi Li (0)) exp χ i ϕ s + χi ⎩ 0 j=m(s)    t 2 |χi | − ds + χi dWs . 2 0

(5.18)

(5.19)

From equation (5.14), the initial conditions are the same for each of the equations (5.16) and (5.19). Then, the right-hand sides of equations (5.16) and (5.19) are equal. Hence, we have equation (5.13) for t ≥ 0. This completes the proof of the first claim.

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Claim 2: The price of each bond becomes 1 at its maturity date. Proof of Claim 2 Since B0 (0) = 1, equation (5.14) implies B1 (0) =

1 . (1 + δ0 L0 (0))

(5.20)

T Substituting i = 1 into equation (5.15), we have B1 (T1 ) = B1 (0) exp{ 0 1 μ ¯(s)ds}. From (5.7), it follows that

 T1  log{1 + δ0 L0 (T0 )} 1 exp ds B1 (T1 ) = 1 + δ0 L0 (0) δ0  0  T1 1 exp log{1 + δ0 L0 (T0 )} . = 1 + δ0 L0 (0) δ0

(5.21)

We see that δ0 = T1 from equation (5.1). From this, we have   δ0 1 exp log{1 + δ0 L0 (0)} B1 (T1 ) = 1 + δ0 L0 (0) δ0 = 1.

(5.22)

Inductively, we obtain Bi (Ti ) = 1 for all i. This proves the second claim. Claim 3: B is arbitrage-free. Proof of Claim 3 Let ξt be an Ito process defined by dξt = −¯ μ(t)dt − ϕt dWt , ξt

(5.23)

with ξ0 = 1. There exists a unique positive solution, given by     t  t |ϕs |2 ds − ϕs dWs . ξt = exp − μ ¯(s) + 2 0 0

(5.24)

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Combining equation (5.15) with equation (5.24), we obtain ⎧ ⎛ i−1 ⎨ t  |ϕs |2 ⎝ ξt Bi (t) = Bi (0) exp − χj ϕ s − ⎩ 0 2 j=m(s) ⎫ ⎛ ⎞ i−1  t i−1 2 ⎬  | j=m(s) χj | ⎝ϕs + − ds − χj ⎠ dWs ⎭ 2 0 j=m(s) ⎧ ⎛ ⎞2 i−1 ⎨  t1  ⎝ϕs + = Bi (0) exp − χj ⎠ ds ⎩ 0 2 j=m(s) ⎫ ⎛ ⎞  t i−1 ⎬  ⎝ ⎠ χj dWs . ϕs + − ⎭ 0

(5.25)

j=m(s)

Hence, ξt Bi (t) is a P-martingale for all i, and ξt is the state price deflator. From these, B is arbitrage-free.

Claim 4: ϕt is well defined as the market price of risk. ¯(t) may be regarded Proof of Claim 4 Since ξt is the state price deflator, μ as the implied short rate from equation (5.23). Indeed, equation (5.7) defines μ ¯(t) as the return rate of the bond with the least time to maturity. We define processes μi (t) and υi (t) by μi (t) = μ ¯(t) − υi (t) = −

i−1 

i−1 

χj (t)ϕt ,

(5.26)

j=m(t)

χj (t),

(5.27)

j=m(t)

respectively. Substituting these into equation (5.12), we obtain dBi (t) = μi (t)dt + υi (t)dWt . Bi (t)

(5.28)

Then, μi (t) and υi (t) indicate the drift coefficient and the volatility, respectively, of the ith bond. From equations (5.26) and (5.27), it follows that μi (t) = μ ¯(t) + υi (t)ϕt ; here, ϕt is independent of the bond i. This coincides

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LIBOR market model (Jamshidian,1997) ȢЍ㻌㻸㼕㻌㻦㻌㼡㼚㼐㼑㼞㻌㻽㻌Ѝ㻌㻮㼚㻌Ѝ㻌㻮㼕㻌Ѝ㻌㻌㻮㼕㻌㻛㻮㼚㻌㻦㻌㻽㻙㻹㻳㻌㻌㻌㻌

LMRW Ȣ㻘㻌ȭ㻌Ѝ㻌㻸㼕㻌㻘㻌㻮㼕㻌㻦㻌㼡㼚㼐㼑㼞㻌㻼㻌Ѝ㻌ȥЍ㻌ȥ㻮㼕㻌㻦㻌㻼㻙㻹㻳㻌㻌㻌㻌

Figure 5.2: Structures to construct the LIBOR market models with the definition of the market price of risk ϕt in Theorem 3.3.1. This completes the proof of the claim and the theorem. ✷ Comparison with Jamshidian’s LIBOR model Let us verify the difference between the LMRW and the Jamshidian model from the viewpoint of arbitrage theory. In Section 5.2, the LIBOR market model is initially constructed as a LIBOR process with given volatility under a measure Q ∼ P. Taking the bond n as a num´eraire, the bond family B is constructed from the LIBOR process. Along these lines, each relative bond price Bi /Bn is a Q-martingale. This approach is illustrated in the upper part of Fig. 5.2. Because of this, there is no information from which to practically derive the state price deflator in this setting, which means that it is not trivial to change the model to a real-world model. In contrast, the LMRW begins with constructing LIBOR and bond price processes such that they are Ito processes represented under the P associated with volatility λ and the market price of risk ϕ. From this, we show that the bond price is consistent with the LIBOR. Next, we define the state price deflator ξt , and so each ξt Bi (t) is a P-martingale. This approach is illustrated in the lower part of Fig. 5.2. By changing the num´eraire, we can derive a risk-neutral model from the LMRW. This will be introduced in Sections 5.4 and 5.5 of this chapter. Arbitrage pricing in the LMRW In the LMRW, we can calculate the state price deflator ξt from equation (5.24).

Libor Market Model

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After this, arbitrage pricing can be executed in the LMRW. For example, consider a security with expiry Ti with payoff at Ti given by a random variable X. The price ct of this security at time t is given by ct = ξ1t Et [ξTi X]. When the expiry T of the security does not coincide with any of Ti , i = 1, · · · , n, the pricing is assumed to be constructed by some technical argument. Resemblance to the BGM model Here, we remark on the resemblance between the LMRW and the BGM model. From equation (5.12), it holds for each i that dBi (t) =μ ¯(t)dt ; t ∈ (Ti−1 , Ti ]. Bi (t)

(5.29)

This indicates that each Bi has null volatility in (Ti−1 , Ti ], the period just prior to its maturity. This assumption is held in common with the setting in the BGM model. Specifically, the BGM model (BGM, 1997, p. 129) assumes the bond price process is given by dB(t, T ) = r(t)dt − σ ˜ (t, T − t)dZt , B(t, T )

(5.30)

where σ ˜ (t, T − t) denotes the bond price volatility, the second variable T − t of σ ˜ represents the time length to the maturity T , and Zt denotes a Brownian motion under the spot measure in the HJM framework. Additionally, the BGM model assumes that σ ˜ (t, T − t) = 0 for t with 0 ≤ T − t < δ. With this, B(t, T ) is represented by dB(t, T ) = r(t)dt ; t ∈ (T − δ, T ]. B(t, T )

(5.31)

This shows the same structure as equation (5.29). Thus, the null volatility of the shortest maturity bond in the LMRW is essentially the same as the assumption on BGM’s volatility σ ˜ (t, T − t). For details, see BGM (1997). 5.4 Spot LIBOR Model This section shows how the LIBOR model under the spot LIBOR measure is implied by the LMRW.

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Num´ eraire From Theorem 5.3.1, we may assume that the LIBOR process Li (t) and the bond price process Bi (t) of the LMRW already exist. Here, we define a process B ∗ (t) for t ∈ [0, τ ] by Bm(t) (t) B (t) = B1 (0)

m(t)−1





(1 + δj Lj (Tj )).

(5.32)

j=1

Note that the process B ∗ (t) did not appear during the construction of the LMRW in Section 5.3. It holds from equation (5.29) that dBm(t) (t) dB ∗ (t) = =μ ¯(t)dt. ∗ B (t) Bm(t) (t) Specifically, B ∗ (t) is expressed by  t  ∗ μ ¯(s)ds . B (t) = exp

(5.33)

(5.34)

0

From this, B ∗ (t) represents a continuous reinvestment in the shortest maturity bonds. Thus, we may adopt the view that B ∗ (t) is a tradable asset. Obviously, it holds that B ∗ (0) = 1 and B ∗ (t) > 0. Given this, we may take B ∗ (t) as a num´eraire. From equation (5.32), it follows that B ∗ (Ti ) = B ∗ (Ti−1 ){1 + δi−1 Li−1 (Ti−1 )}

(5.35)

at each Ti . Since Li−1 (Ti−1 ) indicates the spot LIBOR at t = Ti−1 , B ∗ (t) can be understood as continuous reinvestment at the spot LIBOR. Spot LIBOR measure Combining equations (5.24) and (5.34), we obtain    t  t |ϕs |2 ∗ ϕs dWs . ds − ξt B (t) = exp − 2 0 0

(5.36)

From this, ξt B ∗ (t) is a P-martingale. We define the B ∗ num´eraire measure P∗ by dP∗ = ξ(τ )B ∗ (τ ). dP

(5.37)

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Since B ∗ (t) indicates continuous investment at the spot LIBOR, P∗ is referred to as the spot LIBOR measure. t Setting Zt = 0 ϕs ds + Wt , Zt is a P∗ -Brownian motion, by the Girsanov theorem. Substituting Zt into equations (5.10) and (5.12), we obtain i  dLi (t) = λi (t) χj (t)dt + λi (t)dZt , Li (t)

(5.38)

j=m(t)

i−1  dBi (t) =μ ¯m(t) (t)dt − χj (t)dZt , Bi (t)

(5.39)

j=m(t)

for all i. From these, the bond price process Bi (t) and the LIBOR process Li (t) are specified under the spot LIBOR measure. We call this system a spot LIBOR model to distinguish it from the LMRW. Additionally, we want to verify that the relative bond price is a martingale. Since ξt Bi (t) are P-martingales, each ξt Bi (t)/Et [dP∗ /dP] is a P∗ -martingale, from Proposition 2.4.2. It follows from equation (5.37) that ξt Bi (t) ξt Bi (t) = Et [dP∗ /dP] ξt B ∗ (t) Bi (t) ; i = 1, · · · , n, = B ∗ (t)

a.s.

(5.40)

This means that Bi (t)/B ∗ (t), i = 1, · · · , n are P∗ -martingales. Null market price of risk Here, we examine the case of null market price of risk. When we set ϕt = 0, it then holds that Zt = Wt . From equations (5.24) and (5.34), it holds that ξt = exp{−



t

μ ¯(s)ds} = 0

1 . B ∗ (t)

(5.41)

By equation (5.37), we have dP∗ /dP = ξ(τ )B ∗ (τ ) = 1. Specifically, it holds that P∗ = P almost everywhere. From this, when we assume the null market price of risk on the LMRW, the model is equivalent to the spot LIBOR model. This argument has the same structure as that applied to the HJM model in Section 4.2.

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Pk Measure Model

Let k be a fixed integer with 0 < k ≤ n. This section shows how the LIBOR model under a Bk num´eraire measure is implied from the LMRW. From equation (5.12), the bond price process Bk (t) is given by ⎞ ⎛ k−1 k−1   dBk (t) = ⎝μ ¯(t) − χj ϕ⎠ dt − χj dWt . (5.42) Bk (t) j=m(t)

j=m(t)

The Bk num´eraire measure Pk is defined from equation (3.8) as dPk Bk (τ ) = ξτ . dP Bk (0)

(5.43)

The measure Pk is also referred to as a forward measure, in contrast to the spot LIBOR measure. Of particular note, Pn is referred to as a terminal measure. From equation (5.25), it follows that   k−1 2 t |ϕ + s ξt Bk (t) j=m(s) χj (s)| = exp − ds Bk (0) 2 0 ⎫ ⎛ ⎞  t k−1 ⎬  ⎝ϕs + − χj (s)⎠ dWs . (5.44) ⎭ 0 j=m(s)

We define a process Zk,t by2 ⎛ ⎞  t k−1  ⎝ϕs + Zk,t = χj (s)⎠ ds + Wt . 0

(5.45)

j=m(s)

The process Zk,t is a Pk -Brownian motion, from the Girsanov theorem. Substituting Zk,t into equation (5.10), the LIBOR process Li (t) is represented by k−1  dLi (t) = −λi (t) χj (t)dt + λi (t)dZk,t ; i < k. Li (t) j=i+1

(5.46)

From equation (5.12), the bond price Bi (t) is represented under Pk by ⎫ ⎧ i−1 k−1 i−1 ⎬ ⎨    dBi (t) = μ ¯(t) + χj (t) χk (t) dt − χj (t)dZk,t . (5.47) ⎭ ⎩ Bi (t) j=m(t)

2

k=m(t)

j=m(t)

We denote the Brownian motion under Pk by Zk,t using a subscript k to distinguish it from the lth component of a multi-dimensional Brownian motion Ztl with superscript l.

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With these, the LIBOR model under Pk is implied from the LMRW. We call the resulting system a Pk measure model (or forward measure model) to distinguish it from the spot measure model. Next, we show that the relative bond prices Bi (t)/Bk (t), i = 1, · · · , n are each a Pk -martingale. Since ξt Bk (t) is a P -martingale, we see that   dPk ξt Bk (t) = Et . (5.48) dP Proposition 2.4.2 implies that ξt Bi (t)/Et [dPk /dP] is a Pk -martingale. It follows from equation (5.48) that ξt Bi (t) ξt Bi (t) = Et [dPk /dP] ξt Bk (t) Bi (t) = ; i = 1, · · · , n Bk (t)

a.s.

(5.49)

This means that each of Bi (t)/Bk (t), i = 1, · · · , n is a Pk -martingale. Lognormal distribution Substituting i = k − 1 into equation (5.46), we have dLk−1 (t) = λk−1 (t)dZk,t . Lk−1 (t) From this equation, Lk−1 (t) can be written as   t   t |λk−1 (s)|2 Lk−1 (t) = Lk−1 (0) exp − ds + λk−1 (s)dZk,s . 2 0 0

(5.50)

(5.51)

Since λk−1 (t) is deterministic in t, the forward LIBOR Lk−1 (t) is lognormally distributed under Pk . This is the most important result for option pricing in the LIBOR market model. Remark In derivatives pricing, the forward measure is chosen according to the properties of each derivative. When we use some computer system for derivatives pricing in the LIBOR market model, we typically do not concern ourselves with which measure is used. In practice, such programs typically choose the

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measure in an automatic way according to the properties of each product. For this reason, when we use that model in risk valuation, we have to pay attention to the measure used by the pricing software. Comparison with other measure models As mentioned in Section 4.5, the valuation of the VaR depends on the measure. In this respect, let us compare the value of the LIBOR Li (t) in connection with the choice of measure. In the spot LIBOR model, Li (t) was given in equation (5.38) as i  dLi (t) = λi (t) χj (t)dt + λi (t)dZt , Li (t)

(5.52)

j=m(t)

t where Zt = 0 ϕs ds + Wt . In the Pk measure model, Li (t) was represented in equation (5.46) as k−1  dLi (t) = −λi (t) χj (t)dt + λi (t)dZtl , Li (t) j=i+1

where Zk,t is defined by equation (5.45) as ⎫ ⎧  t⎨ k−1 ⎬  Zk,t = χj (s) ds + Wt . ϕs + ⎭ 0 ⎩

(5.53)

(5.54)

j=m(s)

Specifically, the difference of two Brownian motions is expressed by Zk,t − Zt =



t

k−1 

χj (s)ds.

(5.55)

0 j=m(s)

With this, the numerical difference in Li (t) between the two models is equal to  t  k−1 λi (t) χj (s)ds. (5.56) 0 j=m(s)

Since χj (t) > 0, this difference widens as k or t becomes large. This means that the Pk measure model exhibits a more positive bias in the distribution of Li (t) than the spot LIBOR model does.

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Example 5.5 In practice, a terminal measure Pn is successfully chosen as the risk-neutral measure. To verify the numerical difference between Li (t) as calculated with respect to the measure, let us calculate equation (5.56) in a simple example. We set δ = 0.5 (year) and Ti = δi for i = 1, · · · , 40. We consider the forward LIBOR Li (t) for i = 1, · · · , 39. In this case, the terminal measure P40 indicates the B40 num´eraire measure. Assume a flat term structure, such that the initial LIBOR and the volatility are flat at Li (0) = 0.03 and λi (t) = 0.2 for all i. We have λi (0)δLi (0) 1 + δLi (0) 0.2 × 0.5 × 0.03 = 1 + 0.5 × 0.03 0.003 = 1.015 = 0.002956 ; i = 1, · · · , 39.

χi (0) =

(5.57)

By putting i = 39 and l = 40, the Euler integral of (5.56) gives the difference for L40 (0.5) as λ39 (0)

40−1  i=1

χi (0) × 0.5 = 0.2 × 0.002956 × 39 × 0.5 = 0.01153.

(5.58)

From this, the P40 measure model simulates a LIBOR that is 115 basis points higher at L39 (0.5) (on average) than the spot measure model. This difference strongly affects the computation of VaR. This illustrates that no risk-neutral model is reliable for risk valuation.

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111

Chapter 6 REAL-WORLD MODEL IN THE GAUSSIAN HJM MODEL

Abstract: This chapter theoretically investigates a real-world model within the Gaussian HJM model. In order to construct the real-world model, it is vital to estimate the market price of risk. For this purpose, we assume that the market price of risk is constant during each observation period. Representing the forward rate process in a principal component space, we introduce a formula for the market price of risk as the maximum likelihood estimate. Next, we investigate the numerical properties of the market price of risk, after which we give an interpretation of that price with respect to the historical trend of the forward rates. Furthermore, we show that the interest rate simulation admits historical drift and volatility. Finally, we present a numerical procedure for real-world modeling. These results are essentially those from Yasuoka (2015). Of particular note, however, is that applying maximum likelihood estimation to finding the market price of risk is newly written for this book, in Section 6.2. Additionally, a numerical procedure is introduced in Section 6.9 for implementing the real-world model.

Keywords: Butterfly trading strategy, Dimension reduction, Flattening strategy, Forward rate curve, Full-factor model, Gaussian HJM model, Market price of risk, Monte Carlo simulation, Maximum likelihood estimation, Musiela’s parametrization, Numerical procedure, Observable trend, PCA, Principal volatility component, Real-world simulation, Rolled trend, Rolled trend score, Roll-down, Roll-up, State price deflator, State space, Steepening strategy.

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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F(tk+1, xi-Δt)

Takashi Yasuoka

tk tk+1

F(tk, xi)

tk+2

F(tk+1, xi)

F(tk+2, xi-Δt) T-t=x xi-Δt xi

Time to maturity

Figure 6.1: Forward rate curves and observation of the forward rates There are two main approaches to construct the real-world measure: the socalled forward-looking and backward-looking approaches. A forward-looking approach infers the real-world measure from market information, such as option prices in Ross (2015) (cf. Hull et al.(2014)), which works in the short rate model. A backward-looking approach infers the measure from the historical behavior of forward rates and was proposed by Yasuoka (2013, 2015). The backward-looking approach allows flexible construction of the real-world model since its framework can be applied to a variety of forward rate models, such as the HJM model and the LIBOR market model. This chapter introduces a theory of real-world modeling in the HJM framework following to Yasuoka (2015). 6.1

Discretization of Forward Rate Process

We recall the forward rate process in the HJM model as  t f (t, T ) = f (0, T ) + {−σ(s, T )υ(s, T ) + σ(s, T )ϕs }ds 0  t + σ(s, T )dWs ,

(6.1)

0

where Wt is a d-dimensional P-Brownian motion. The HJM model is called Gaussian if σ(t, T ) is a deterministic function of t and T . In this book, we

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always assume that volatility σ(t, T ) is deterministic and continuous with respect to t and T . Then, υ(t, T ) is also deterministic and continuous. In the following, the market price of risk is assumed to be constant. The validity of the constancy assumption will be examined in the context of risk management in Chapter 7. Data observation Denoting the market price of risk as a constant ϕ, the above forward rate process is expressed by  t f (t, T ) = f (0, T ) + {−σ(s, T )υ(s, T ) + σ(s, T )ϕ}ds 0  t + σ(s, T )dWs . (6.2) 0

Next, we specify a historical dataset as follows. Let a time interval Δt > 0 be fixed, and {tk }k=1,··· ,J+1 be a sequence of observation dates such that t1 = 0 and tk+1 − tk = Δt, where J + 1 is the number of observation times. We denote the time length to a maturity T from t by x = T − t. For an integer n ≥ d, x1 , · · · , xn denotes a sequence of time lengths to maturity. Typically, we observe the instantaneous forward rate F (tk , xi ) with respect to fixed xi . Fig. 6.1 illustrates an example of forward rate curves observed at tk ,tk+1 , and tk+2 , showing F (tk , xi ), F (tk+1 , xi − Δt), and so on. We assume that the dynamics of these observations follow equation (6.2). We substitute F (tk , xi ) and F (tk+1 , xi − Δt) into f (0, xi ) and f (Δt, xi ) , respectively, in equation (6.2) for every tk , k = 1, · · · , J. Then, it follows that  Δt {−σ(s, xi )υ(s, xi ) F (tk+1 , xi − Δt) = F (tk , xi ) + 0  Δt +σ(s, xi )ϕ}ds + σ(s, xi )dWs (6.3) 0

for i = 1, · · · , n and k = 1, · · · , J, where xi is identified with a maturity date at t = 0. Since σ is assumed to be continuous, the Euler integral implies

F (tk+1 , xi − Δt) = F (tk , xi ) + {−σ(0, xi )υ(0, xi ) + σ(0, xi )ϕ}Δt √ + Δt σ0i W1 (6.4)

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for all i and k, where W1 = (W11 , · · · , W1d )T corresponds to a d-dimensional standard normal distribution. To simplify the notation, we write σ(0, xi ) 1 d T and υ(0, xi ) as σ0i and υ0i , respectively, where σ0i = (σ0i , · · · , σ0i ) and υ0i = 1 d T (υ0i , · · · , υ0i ) . This yields a kind of regression form, as √ F (tk+1 , xi − Δt) = F (tk , xi ) + {−σ0i υ0i + σ0i ϕ}Δt + Δt σ0i W1 , (6.5) for all i and k. Remark When we put t = k + 1 into the above, we obtain the next observation, which occurs at time tk+1 , as √ F (tk+2 , xi − Δt) = F (tk+1 , xi ) + {−σ0i υ0i + σ0i ϕ}Δt + Δt σ0i W1 . (6.6) The left-hand side of equation (6.5) does not appear in the right-hand side of the above equation because F (tk+1 , xi − Δt) = F (tk+1 , xi ). Thus, the formulation (6.5) is slightly different from the usual regression model that appears in Appendix C. 6.2

Estimation of Market Price of Risk

We recall the volatility structure associated with PCA in Section 4.3. A sample covariance matrix V is defined by Vij =

1 Cov(F (tk + Δt, xi − Δt) − F (tk , xi ), Δt F (tk + Δt, xj − Δt) − F (tk , xj )) ; i, j = 1, · · · , n.

(6.7)

We assume that V has rank d ≤ n. By the argument in Appendix B, the  covariance matrix is decomposed into Vij = dl=1 eli ρ2l elj for i, j ≤ n, where ρ2l is the lth eigenvalue, and el = (el1 , · · · , eln )T is the lth principal component of the covariance for l = 1, · · · , d. We always assume that el1 > 0,

ρl > 0 ; l = 1, · · · , d.

(6.8)

This assumption is significant in the interpretation of the meaning of the market price of risk in Sections 6.4 and 6.5. Recall the equation (B.4), that is,

Gaussian HJM Model

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that the principal components e1 , · · · , ed form an orthonormal set. Thus, n  i=1

eli ehi = δlh ; l, h = 1, · · · , d.

(6.9)

Representation in principal component space Next we represent the dynamics of the forward rates in the principal component space. We suppose that the volatility is given by the principal component as l = ρl eli ; l = 1, · · · , d, i = 1, · · · , n. σ0i

(6.10)

Here, σ l is the lth volatility component. From equation (4.36), we see that |σ l |2 = (ρl )2 , and so ρl represents the magnitude of the lth volatility component σl . With respect to the sequence {xi }i=1,··· ,n , we assume that the rank of the n × d-matrix ⎞ ⎛ 1 d σ01 · · · σ01 ⎜ .. .. ⎟ (σ 1 , · · · , σ d ) = ⎝ ... (6.11) . . ⎠ 1 d σ0n · · · σ0n

is equal to d. Next, we set ΔFi (tk ) by letting ΔFi (tk ) = F (tk+1 , xi − Δt) − F (tk , xi ).

(6.12)

With this, equation (6.5) becomes √ ΔFi (tk ) = {−σ0i υ0i + σ0i ϕ}Δt + Δt σ0i W1 d d  √  l l σ0i W1l , = −σ0i υ0i Δt + σ0i ϕl Δt + Δt l=1

(6.13)

l=1

where the left-hand side represents an observation and the right-hand side represents a random variable. Let us consider a d-dimensional principal component space with a basis 1 e , · · · , ed . Regarding 

ΔFn (tk ) ΔF1 (tk ) ,··· , Δt Δt

T

(6.14)

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as an n-dimensional vector in Rn , we denote by αl (tk ) the inner product1 of the above with el as αl (tk ) =

n  ΔFi (tk ) i=1

Δt

eli .

(6.15)

In other words, αl (tk ) represents the lth projection of the vector (6.14). Additionally, we define a constant βl by βl = =

n 

σ0i υ0i eli

i=1 ! d n   i=1

(6.16)

l l σ0i υ0i

eli ; l = 1, · · · , d.

l=1

From equation (6.9), the lth projection of equation (6.13) is represented as αl (tk )Δt =

n 

ΔFi (tk )eli

i=1

= −

n 

σ0i υ0i Δteli + n 

d 

m σ0i W1m eli

i=1 m=1 d 

= −βl Δt + √ + Δt

m σ0i ϕm Δteli

i=1 m=1

i=1

√ + Δt

n  d 

ρm ϕm

m=1

d 

ρm W1m

m=1

= (−βl + ρl ϕl )Δt +

n 

l em i ei Δt

i=1 n 

l em i ei

i=1



Δtρl W1l

(6.17)

at all observation times tk , k = 1, · · · , J. This is an expression of equation (6.13) in the space spanned by the principal components e1 , · · · , ed . We remark again that the left-hand side is an observation at time tk , and the right-hand side is represented by a random variable. 1

The α in the following is not related to the α of Chapter 4.

Gaussian HJM Model

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Maximum likelihood estimation of the market price of risk The last term in the right-hand side of equation (6.17) is normally distributed. From Proposition C.0.1, the maximum likelihood estimate of ϕl is given by the solution to the least squares problem with minimization of the following: J  k=1

|αl (tk )Δt + (βl − ρl ϕl )Δt|2 .

(6.18)

Alternatively, we may reduce the above to the problem of minimizing θl (ϕl ) with J  θl (ϕl ) = |αl (tk ) + (βl − ρl ϕl )|2 . (6.19) k=1

This follows because Δt is a non-zero constant. Since each θl (ϕl ) is a onevariable function of ϕl , it is sufficient to solve d minimization problems independently, one for each of θl (ϕl ), l = 1, · · · , d. Then, the PCA setup simplifies maximum likelihood estimation. Although the PCA formulation is indirect, this setup provides us with a perspective on estimation of the market price of risk. As a result, the next theorem gives an exact formulation of the market price of risk, framing it as the solution to the minimization problem. Theorem 6.2.1 For a Gaussian HJM model, if the volatility σ(t, T ) is continuous on t and T , and σ l is given as the lth volatility component for each l, then the maximum likelihood estimate of the market price of risk ϕl is uniquely given by # 1 " H (6.20) E [αl ] + βl ; l = 1, · · · , n, ϕl = ρl n n 1 H l l where αl (tk ) = Δt i=1 σ0i υ0i ei , and E [ ] denotes a i=1 ΔFi (tk )ei , βl = sample mean of the historical data for t1 , · · · , tJ . Proof

Direct calculation implies

J  ∂θ(ϕl ) = −2 {αl (tk ) + (βl − ρl ϕl )}ρl ∂ϕl k=1

(6.21)

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and ∂ 2θ = 2(ρl )2 J. ∂ϕ2l

(6.22)

From this, θl (ϕl ) is strictly convex with respect to ϕl for all l because of the assumption ρl > 0. Therefore, the least squares solution exists uniquely as the root of the equation such that ∂θ(ϕl )/∂ϕl = 0. From equation (6.21) and ρl = 0, we may solve the following equation: J  {αl (tk ) + (βl − ρl ϕl )} = 0. (6.23) k=1

Denoting by E H [ ] the sample mean for the observation t1 , · · · , tJ , we have  E H [αl ] = J1 Jk=1 αl (tk ). Substituting this into equation (6.23), it follows that ρl ϕ l

J 1 = αl (tk ) + βl J k=1

= E H [αl ] + βl .

(6.24)

# " Consequently, ϕl is solved as ϕl = ρ1l E H [αl ] + βl . This completes the proof. The exact formulation (6.20) for the market price of risk enables two significant applications. First, it becomes possible to systematically study the properties of the market price of risk. Second, the properties of the interest rate simulation can be theoretically examined, which will be described in Sections 6.5, 6.6, and 6.7. However, in this approach, we have not dealt with the t-value for the statistics in the market price of risk estimation; this is a further subject. Remark There is another approach to represent the dynamics of the forward rates g(t, x), applying the so-called Musiela’s parametrization as   ∂ dg(t, x) = g(t, x) + a(t, x) + σ ˜ (t, x)dWt , (6.25) ∂x

Gaussian HJM Model

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where x = T − t is the time length from time t to maturity T . For details, see Brace and Musiela (1994). Usually, we observe the interest rate with respect to the variable x, and so the above equation might be more suitable for specifying historical data on the forward rates than the HJM model is. However, because we are considering the real-world model as our primary focus, the difference between these two approaches seems less important in the beginning. Therefore, we employ the simple representation in the HJM model. 6.3

Market Price of Risk: State Space Setup

This section introduces another method to estimate the market price of risk: working in a state space. Denoting the market price of risk by ϕ′ = (ϕ′1 , · · · , ϕ′d )T , we return to the discretization as equation (6.13), which we reproduce below: √ ΔFi (tk ) = −σ0i υ0i Δt + σ0i ϕ′ Δt + Δt σ0i W1 ; i = 1, · · · , n, k = 1, · · · , J.

(6.26) We remark that the volatility is assumed to be determined by a principal component. Our objective here is to directly obtain ϕ′ from the above equations. We denote by ǫ(ϕ′ ) the sum of the squared difference between each side of equation (6.26) in the time series and cross sections, neglecting the random part, such that J n 1  2 ǫ(ϕ ) = {ΔFi (tk ) + (σ0i υ0i − σ0i ϕ′ )Δt} . J k=1 i=1 ′

(6.27)

Let ϕ′ be the solution that minimizes ǫ(ϕ′ ). We call this setting a state space setup, and call that used in the previous section a PCA setup to distinguish between the two approaches. We note the implications of both definitions below. • ϕ is the solution that minimizes θl (ϕl ) in equation (6.19) in the principal component space, and also is the maximum likelihood estimate. • ϕ′ is the solution that minimizes ǫ(ϕ′ ) of equation (6.27) in the state space.

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Note that the definition of ǫ(ϕ′ ) does not deal with the PCA. Then, the derivation of ǫ(ϕ′ ) is immediate and more comprehensive than that of θ(ϕ). Accordingly, the definition of ϕ′ is simpler than that of ϕ. However, the minimization problem (6.27) is too complicated to allow deriving ϕ′ by maximum likelihood estimation. Apart from this, let us solve this least squares problem. We have the following result. Proposition 6.3.1 For a d-dimensional discretization of equation (6.26) with d ≤ n, let ǫ(ϕ′ ) be a function of d variables in ϕ′1 , · · · , ϕ′d and defined by equation (6.27). Then, the solution ϕ′ that minimizes ǫ(ϕ′ ) is uniquely given as the solution of the system of linear equations, as ! n d n    m l l σ0i ϕ′m = σ0i γi σ0i ; l = 1, · · · , d, (6.28) m=1

i=1

i=1

where γi = E

H



 ΔFi + σ0i υ0i ; i = 1, · · · , n. Δt

We may assume that the matrix ⎞ ⎛ 1 d σ01 · · · σ01 ⎜ .. ⎟ .. (σ 1 , · · · , σ d ) = ⎝ ... . ⎠ .

(6.29)

Proof

1 σ0n

has rank d. From σ0i ϕ′ = ∂σ0i ϕ′ l = σ0i . ′ ∂ϕl

···

d

l=1

(6.30)

d σ0n

l σ0i ϕ′l , it follows that

(6.31)

Then, the partial derivative of ǫ(ϕ′ ) in ϕ′l is J

n

∂ǫ(ϕ′ ) −2   l = {ΔFi (tk ) + (σ0i υ0i − σ0i ϕ′ )Δt} σ0i Δt ′ ∂ϕl J k=1 i=1 n  " H # l E [ΔFi ] + (σ0i υ0i − σ0i ϕ′ ) Δt σ0i Δt = −2 i=1

(6.32)

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HJM Model

for l = 1, · · · , d. From equations (6.31) and (6.32), we have n

 ∂ 2 ǫ(ϕ′ ) l m = 2 σ0i σ0i Δt2 . ′ ∂ϕl ∂ϕ′m i=1

(6.33)

From this, the Hessian matrix of ǫ(ϕ′ ) is represented by ∇2 ǫ(ϕ′ ) = 2Δt2 {σ 1 , · · · , σ d }T {σ 1 , · · · , σ d }.

(6.34)

It holds that for an arbitrary vector y( = 0) ∈ Rd , y T {σ 1 , · · · , σ d }T {σ 1 , · · · , σ d }y = |{σ 1 , · · · , σ d }y|2 > 0,

(6.35)

because the rank of {σ01 , · · · , σ0n } is d. That is, ∇2 ǫ(ϕ′ ) is positive definite. Hence, the solution ϕ′ to the minimization problem exists uniquely and is obtained by solving the equation ∇ǫ(ϕ′ ) = 0. For a simple expression, we define a constant vector γ = (γ1 , · · · , γn )T by setting   H ΔFi + σ0i υ0i . (6.36) γi = E Δt Note that γ is completely determined by the historical data, and specifically by the forward rates and the volatility. Upon substituting equation (6.36) into (6.32), it should hold that n

 ∂ǫ(ϕ′ ) l = −2 (γi − σ0i ϕ′ )σ0i Δt2 = 0. ∂ϕ′l i=1

(6.37)

This can be reduced to n  i=1

l (γi − σ0i ϕ′ )σ0i = 0 ; l = 1, · · · , d.

(6.38)

By direct calculation, n 

l σ0i ϕ′ σ0i =

i=1

=

n 

i=1 d 

m=1



d 

n  i=1

l σ0i



ϕ′m .

m ′ σ0i ϕm

m=1





m l σ0i σ0i

(6.39)

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Substituting this into equation (6.38), we obtain the system of linear equations on ϕ′ ,   n d n    m l ′ l σ0i σ0i ϕm = ; l = 1, · · · , d. (6.40) γi σ0i m=1

i=1

i=1

"n # m l ′ Since the d × d matrix i=1 σ0i σ0i ml is positive definite and has rank d, ϕ is obtained as a unique solution to equation (6.40). This completes the proof. Proposition 6.3.1 does not assume orthogonality within the volatility structure. This means that the result is available for an artificially given volatility; for example, it can be applied with a Nelson–Siegel type volatility, Hull–White type volatility, and so forth. Furthermore, this proposition is applicable to a kind of a Markovian model studied by Ritchken and Sankarasubramanian (1995), being available whenever the volatility is deterministic and continuous. Principal volatility component We remark that Proposition 6.3.1 does not involve the maximum likelihood estimate for the market price of risk. To change that, we additionally suppose that the volatility is given by the principal volatility components as l σ0i = ρl eli ; l = 1, · · · , d, I = 1, · · · , n,

(6.41)

where ρ2l (> 0) is the lth eigenvalue, and el = (el1 , · · · , eln ) is the lth principal component of the covariance. We reproduce the least squares problem (6.27) in the state space as n J 1  2 {ΔFi (tk ) + (σ0i υ0i − σ0i ϕ′ )Δt} . ǫ(ϕ ) = J k=1 i=1 ′

(6.42)

We have the following theorem. Theorem 6.3.2 For a Gaussian HJM model, we assume that the volatility σ(t, T ) is continuous on t and T , and the values of σ0i are given as the principal components by equation (6.41). Then, ϕ′l is uniquely given by   n 1  H ΔFi ′ {E (6.43) + σ0i υ0i }eli ; l = 1, · · · , d. ϕl = ρl i=1 Δt

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Proof From Proposition 6.3.1, we have   n n d    l m l ′ γi σ0i . σ0i σ0i ϕm =

(6.44)

From equation (6.41), the left-hand side of the above becomes     n d d n     m l l σ0i σ0i ϕ′m = ϕ′m . em ρm ρl i ei

(6.45)

m=1

m=1

i=1

i=1

m=1

i=1

i=1

By the orthonormality of the principal components, the above equation can be reduced to   n d d    m l ′ σ0i σ0i ϕm = ρm ρl ϕ′m δml m=1

i=1

=

m=1 (ρl )2 ϕ′l .

(6.46)

Similarly, the right-hand side of equation (6.44) can be written as n  i=1

l = ρl γi σ0i

n 

γi eli .

(6.47)

i=1

Since equations (6.46) and (6.47) are the same, it follows that n 1  l ′ ϕl = γ i ei , ρl i=1

(6.48)

where γi was given by equation (6.29) as γi = E H [ΔFi /Δt] + σ0i υ0i . Substituting this into equation (6.48), ϕ′l is represented by equation (6.43). This completes the proof. ✷ Accordingly, we have the following corollary, which shows that the market price of risk derived in the state space setup is implied as the maximum likelihood estimate. Corollary 6.3.3 Under the same conditions as in Theorem 6.3.2, it holds that ϕ′l = ϕl ; l = 1, · · · , d.

(6.49)

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F(t1,・)

Observable trend > 0

Observable trend < 0

F(t1,・)

F(tJ ,・)

Term to maturity

0 x1 x 2

xn

(a) Positive observable trend

Term to maturity

0 x1 x2

xn

(b) Negative observable trend

Figure 6.2: Change in the forward rates and the observable trend Proof

From equation (6.20) of Theorem 6.2.1, ϕl is given by

# 1 " H (6.50) E [αl ] + βl , ρl n n 1 l l where αl (tk ) = Δt i=1 σ0i υ0i ei . Substituting these i=1 ΔFi (tk )ei and βl = into equation (6.50), we obtain ϕl =

   n  1  H ΔFi (tk ) ϕl = + σ0i υ0i eli = ϕ′l . E ρl i=1 Δt

(6.51)

This completes the proof. 6.4

Historical Trend of Forward Rate

By using Theorem 6.2.1, we investigate the numerical implications of the market price of risk with respect to the historical change in the forward rates. For this purpose, we provide some definitions to use in describing this change. Note that approximations and deductions here have been inferred from empirical observation. Observable trend Let tk , k = 1, · · · , J + 1 be a sequence of observation dates of the forward rates F (tk , xi ) such that tk+1 − tk = Δt > 0. The observable trend of the historical

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forward rate F (·, xi ) is defined by   H F (tk+1 , xi ) − F (tk , xi ) E , Δt

(6.52)

where E H [ ] denotes the sample mean. We see that   1 H F (tk+1 , xi ) − F (tk , xi ) E = [{F (t2 , xi ) − F (t1 , xi )} Δt ΔtJ + · · · + {F (tJ+1 , xi ) − F (tJ , xi )}] 1 = {F (tJ+1 , xi ) − F (t1 , xi )}. (6.53) ΔtJ From this, the observable trend for xi is positive when F (tJ+1 , xi ) − F (t1 , xi ) > 0. Fig. 6.2(a) illustrates a case in which the observable trends are positive for all xi . Conversely, this trend is negative when F (tJ+1 , xi ) − F (t1 , xi ) < 0, as shown in Fig. 6.2(b), which illustrates a case in which observable trends are negative for all xi . Thus, the observable trend represents the average change in the forward rate F (·, xi ) for a fixed xj during the sample period [t1 , tJ+1 ]. More precisely, this trend represents the average speed of change in the forward rate. Rolled trend The observable trend is visually comprehensive, as shown in Fig. 6.2. Although the term “observable trend” is newly coined for our investigation, this measure is common in the analysis of market interest rates. So, knowing this trend is not sufficient for exactly investigating the implications of the market price of risk. To supplement this trend, we introduce the rolled trend, as follows. The rolled trend of the historical forward rate F (·, xi ) is defined for fixed xi as   H ΔFi E , (6.54) Δt where ΔFi (tk ) = F (tk+1 , xi − Δt) − F (tk , xi ). It follows that     H ΔFi (tk ) H F (tk+1 , xi ) − F (tk , xi ) E = E Δt Δt   H F (tk+1 , xi ) − F (tk+1 , xi − Δt) . −E Δt

(6.55)

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F(tJ ,・)

Roll up

F(t1,・)

0 x1 x2

Term to maturity xn

(a) Negative rolled trend

Takashi Yasuoka

F(tJ ,・) F(t1,・) Term to maturity

0 x1 x2

xn

(b) Positive rolled trend

Figure 6.3: Rolled trend and roll-up/down of the forward rates The first term is equal to the observable trend of F ( , xi ), and the second term corresponds to the historical mean of the slope of the forward rate at x = xi . Roll down and roll up For example, let us consider a case in which the forward rate curve has positive slope and rises slowly, as in Fig. 6.3(a). In this case, the first term of equation (6.55) is nearly equal to 0, and the second term is negative because   H F (tk+1 , xi ) − F (tk+1 , xi − Δt) E > 0. (6.56) Δt This case implies that the rolled trend is negative, as E H [ΔFi /Δt] < 0, which is known as the roll-down of the forward rate curve in practice. When the upward-sloping forward rate curve rises quickly across its entirety, the curve rolls up, as shown in Fig. 6.3(b). Although the second term might be positive, the first term is both positive and larger than the second term. As a result, the rolled trend might be positive in this case. Thus, the rolled trend corresponds to the roll-down or roll-up of the forward rate curve, with which we are familiar from financial experience. lth trend score Next, we represent both trends in the principal component space.

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Gaussian HJM Model

F(tJ ,・)

F(tJ ,・)

F(t1,・)

Flattening

F(tJ ,・)

F(t1,・)

F(tJ ,・)

Steepening Term to maturity

Term to maturity 0 x 1 x2

xn

0 x1 x 2

(a) Second score

xn (b) Third score

Figure 6.4: Second and third score of rolled trend Definition 6.4 For l = 1, · · · , d, the lth observable trend score Ol is defined by   n  H F (tk+1 , xi ) − F (tk , xi ) Ol = eli . (6.57) E Δt i=1 The lth rolled trend score Rl is defined by   n  H ΔFi E Rl = eli . Δt i=1

(6.58)

The following lemma is directly obtained from equations (6.15) and (6.58). Lemma 6.4.1 Rl = E H [αl ] ; l = 1, · · · , d

(6.59)

Meaning of R1 As mentioned in Section 4.3, the first principal component represents a parallel shift, with an example shown in Fig. 6.2. If the entire forward rate curve with positive-slope rises slowly, like that shown in Fig. 6.3(a), then the first rolled trend score is negative, that is, R1 < 0. In contrast, if the entire forward rate curve rises fast, as shown in Fig. 6.3(b), then it follows that R1 > 0. Table 6.1 roughly shows the value of R1 with respect to the shape of the forward rate curve and the observable trend. For example, the value of the

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latter case above is exhibited in the cell in the “Upward sloping” row and “Rising fast” column, where R1 is positive. Table 6.1 Qualitative estimation of the first rolled trend R1

Shape of forward rate curve Upward-sloping Flat Downward-sloping

Observable trend Falling fast Stable Rising fast Negative Negative Positive Negative Near zero Positive Negative Positive Positive

Meaning of R2 and R3 Next, we examine the meaning of the second rolled trend. We assumed that el1 > 0 in equation (6.8) for all l, which affects the sign of the score. We suppose that the forward rate curve is upward sloping and steepening during the sample period, as illustrated by the dashed line in Fig. 6.4(a). This change is opposite to the second component of the forward rates at Fig. 6.2 in Section 4.3. Then, the first rolled trend score might be negative, and the second rolled score is negative; that is, R1 < 0 and R2 < 0. Conversely, if the forward rate curve is flattening, illustrated by the narrow line in Fig. 6.4(a), then the first rolled trend score might be negative, and the second one positive; we then see that R1 < 0 and R2 > 0. Accordingly, Table 6.2 qualitatively estimates the values R1 and R2 with respect to the observable trend scores O1 and O2 . The former example in the above is shown in the row marked “Steep.” The latter example is shown in the row marked “Flat.” The explanation for the third rolled trend score is similar to the explanations above. Summarizing the above argument, O1 , O2 , and O3 represent the level, slope, and curvature factors of the observed trend. Similarly, R1 , R2 , and R3 represent the same factors of the rolled trend. Remark In Table 6.2, “bull” means that the forward rate curve is falling, and “bear” means it is rising. Additionally, “steepening” means the slope of the forward rate curve is become larger, with “flattening” meaning the slope is becoming

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smaller. “Bull steepening” is a combination of the bull and steepening changes in the forward rate curve, and other combinations are analogously defined. Table 6.2 Qualitative estimation of R1 and R2 with respect to the observable trends. Forward rate curves are assumed to be upward sloping during the sample period.

Observable trend Bull steep Steep Bear steep Bull flat Flat Bear flat 6.5

R1 Negative Negative Near zero Negative Negative Near zero

R2 Negative Negative Negative Positive Positive Positive

Market Price of Risk and the Trends

This section briefly examines the numerical meaning of the market price of risk in connection with the rolled trend. For this, Theorem 6.2.1 makes it possible to derive a qualitative estimate of the value of the market price of risk. Approximation of the market price of risk The next proposition shows that the lth market price of risk is approximately explained as the ratio of the lth rolled trend score to the magnitude of the lth volatility component. The argument here is developed in the principal component space. Proposition 6.5.1 For a d-dimensional Gaussian HJM model, we assume that the volatility σ(t, T ) is continuous on t and T , and σ0i is given as the principal component by equation (6.10). If the inequality   H ΔFi (6.60) |σ0i υ0i | ≪ E Δt holds for all i, then the lth market price of risk is approximated by ϕl ≈

Rl ; l = 1, · · · , d, ρl

(6.61)

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where Rl is the lth rolled trend score, and ρl is the magnitude of the lth volatility component. Proof |

From equation (6.60), we have n  i=1

σ0i υ0i eli |



n 

E

i=1

H



 ΔFi l ei Δt

(6.62)

for all l. Substituting equations (6.15) and (6.16) into the above, it follows that   n  H ΔFi |βi | ≪ eli E Δt i=1 = E H [αl ] .

(6.63)

By Theorem 6.2.1, we have # 1 " H E [αl ] + βl ρl 1 H ≈ E [αl ] . ρl

ϕl =

(6.64)

From equation (6.59) in Lemma 6.4.1, it holds for all l that Rl = E H [αl ]. Substituting this into equation (6.64), we obtain equation (6.61). This completes the proof. A remark on inequality (6.60) We want to note here that the condition (6.60) is empirically feasible. From the definition of υ in equation (4.10), we have, roughly,  Ti υ0i (= υ(0, Ti )) = − σ(0, u)du 0

≈ σ(0, Ti )Ti = σ0i Ti .

(6.65)

l l l 2 Accordingly, it holds that |σ0i υ0i | ≈ |σ0i | Ti for each l. Then, if the volatility σ is uniformly low in i, the left-hand side value of equation (6.60) may be assumed to be sufficiently small.

Gaussian HJM Model

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Additionally when we work with a period where the interest rate rises (or falls) a consider amount, we may expect that the following will hold: E H [ΔFi /Δt] ≫ 0. Therefore, when the volatility is sufficiently low, the condition |σ0i υ0i | ≪ E H [ΔFi /Δt] can be satisfied. The approximation (6.61) is practically useful for investigating the implications of the market price of risk. Of course, we are not asserting that the above inequality always holds. Qualitative estimation for ϕ1 Proposition 6.5.1 states that the market price of risk is largely determined by the rolled trend in the principal component space. Recall that R1 , R2 , and R3 represent the level, slope, and curvature factors of the rolled trend, respectively. Since ρl represents the magnitude of the lth volatility component, we may take the view that ρl indicates the “lth volatility risk.” Hence, we may say that lth market price of risk is approximately equal to the magnitude of the lth rolled trend per unit risk of the lth volatility component; that is, {lth market price of risk} ≈

{Magnitude of lth rolled trend} . {lth volatility risk}

(6.66)

Accordingly, this interpretation can be roughly rephrased in terms of the observable trend. In particular, for l = 1, from equation (6.55) we may accept that R1 ≈ O1 − {Slope of forward rate curve} (6.67) holds approximately. Combining equations (6.61) with (6.67), Table 6.3 presents a rough estimate of ϕ1 in terms of the observable trend, where each result corresponds to the relevant entry in Table 6.1. As an example, suppose that the forward rate curve is upward-sloping and stably unchanged, as shown in Fig. 6.5(a). Then, O1 is almost equal to zero and R1 is negative; that is, this is the roll-down. Hence, ϕ1 might be negative. Moreover, if the forward rate curve rises fast as shown in Fig. 6.5(b), such that O1 is positive and larger than the second term in equation (6.67), then R1 is positive. Hence, ϕ1 might be positive. These two cases are shown in the first row of Table 6.3. Following this line of reasoning, ϕ1 is roughly estimated for other shapes of the forward rate curve. If the curve is almost flat during the sample period,

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F(tJ ,・)

Roll up

Takashi Yasuoka

F(tJ ,・)

F(t1,・)

F(t1,・) Term to maturity

Term to maturity 0 x1 x 2

0 x1 x2

xn

(a) O1 is near zero; R1 is negative.

xn

(b) O1 and R1 are positive.

Figure 6.5: First observable trend score and rolled trend then the second term in equation (6.67) is nearly equal to zero. We may assume that O1 ≈ R1 . Therefore, ϕ1 is almost estimated by the observable trend, which is shown in the second row of Table 6.3. In Section 7.2, we will work with this particular case for studying the negative price tendency of ϕ1 . An example with actual data will be presented in Section 10.4. At last, we examine the case in which the forward rate curve is downwardsloping. This means that the second term in equation (6.67) is negative, and so it roughly holds that R1 > O1 . With this, ϕ1 is estimated to be a bit larger than in the case of a flat curve, which is shown in the bottom row of Table 6.3. Table 6.3 First market price of risk ϕ1 with respect to the observable trend and average shape of the forward rate curve.

Shape of forward rate curve Upward-sloping Flat Downward-sloping

Observable trend Fast fall Stable Fast rise Negative Negative Positive Negative Near zero Positive Negative Positive Positive

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Table 6.4 First and second market prices of risk with respect to the observable trend. The forward rate curve is assumed to be upward-sloping through a sample period.

Observable trend Bull steep Steep Bear steep Bull flat Flat Bear flat

ϕ1 Negative Negative Near zero Negative Negative Near zero

ϕ2 Negative Negative Ne0ative Positive Positive Positive

Qualitative estimation for ϕ2 Next, we examine the properties of the first and second market price of risk ϕ1 and ϕ2 in connection with R1 and R2 , respectively. We assume that the forward rate curve is almost upward-sloping through the observation period. We have roughly estimated the rolled trend score R1 and R2 for each observable trend in Table 6.2. From those estimations, the market prices of risk ϕ1 and ϕ2 are roughly inferred by Proposition 6.5.1, with the results summarized in Table 6.4. For example, the steepening of the forward rates implies that O1 ≈ 0 (or ≤ 0) and O2 < 0, as shown in Fig. 6.6, Then, O1 is near zero and R1 might be a little negative because of the roll-down effect. Hence, ϕ1 might be a little negative. Since O2 is negative, R2 might be negative, and so ϕ2 takes a negative value. This case is shown in the second row of Table 6.4. Furthermore, when the forward rate curve is flat or has negative slope, we can estimate the market price of risk by a method similar to the above.

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F(tJ ,・) F(t1,・) Steepening Term to maturity 0 x1 x2

xn

O1 ≈ 0, O2 < 0 and R1 < 0, R2 < 0

Figure 6.6: Observable trend and rolled trend. The forward rate curve is steepening through a sample period.

6.6

Interpretation of Market Price of Risk

Recall the conventional argument on the negative market price of risk, discussed in Section 4.2. From equation (4.22), the market price of risk satisfies μi (t) − r(t) = υ(t)ϕt for all Ti . This means that ϕt is independent from the maturity Ti . In the one-dimensional case, the above equation is reduced to ϕt =

μi (t) − r(t) . υ(t)

(6.68)

From this form, the market price of risk is traditionally explained as the excess return per unit risk for investment in the ith bond; that is, it is the same risk– return no matter which bond we invest in within an arbitrage-free market. The traditional interpretation is familiar within the usual investment theory, but it is difficult to develop along these lines for the higher-order market price of risk. In this respect, this section presents another interpretation of the market price of risk, employing Proposition 6.5.1 to do so. We begin with addressing the roll-down return from bond investment. Roll-down return In order to explain the roll-down effect in the context of bond investment, we

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Gaussian HJM Model

2 years elapse 5% 4%

Yield curve Roll down Maturity 0

1

2

3

4

5

6

7

Figure 6.7: Yield curve and roll-down work with the yield rates instead of the forward rates for a while. We assume that the yield curve has positive slope, as shown in Fig. 6.7. Recall the five-year bond with coupon rate 5% in Example 1.4.1. From calculation with equation (1.29), we can buy that bond at the face value 1.00. Two years later, we will own a three-year bond with coupon rate 5%. If the yield curve has not changed, then the yield of the three-year bond rolls down to 4%, as shown in Fig. 6.7. This rate is taken as the discount rate, as follows. Specifically, the price of this bond is equal to the present value of the three-year bond with 5% coupon by applying the discount rate of 4%. From the calculation of equation (1.34), the price rises to 1.02775; that is, we gain 0.02775 (= 1.02775 − 1.00) by the roll-down. Therefore, the roll-down on the yield provides excess return for a bond holder. Conversely, if the three-year yield rises to 6% after two years, that will be a roll-up of the yield. Then, the price of the bond will fall to 0.9733 from the calculation of equation (1.36). We then lose 0.0267 (= 0.9733 − 1.00) due to the roll-up. Therefore, when the yield curve is steep, the bond holder can expect an excess return from the roll-down. The roll-down and roll-up correspond to excess return and loss, respectively, for the investor. These observations approximately follow the analogous observations for the roll-down and roll-up in the forward rates. Now we return to the argument on forward rates.

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Loss Profit Portfolio Maturity 0

1

2

3

4

5

6

(a) Uniform portfolio

Loss Profit Portfolio Maturity 0

1

2

3

4

5

6

(b) Sell long-term bonds, and buy short-term bonds

Loss Profit

Portfolio 0

1

2

3

4

Maturity 5

6

(c) Buy short- and long-term bonds, and sell middle-term bonds

Figure 6.8: Risk–return in three investment strategies

Takashi Yasuoka

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The first market price of risk for investor Now we explain the first market price of risk in connection with the roll-down return by referring to Fig. 6.8. The bar graph in the lower part of Fig. 6.8(a) shows a uniform portfolio consisting of one-year to six-year bonds. The investor expects the roll-down of the entire forward rate curve for the excess return. Specifically, R1 < 0

(6.69)

is expected. The risk for the investor is the roll-up of the entire forward rates, in other words, the level change of the rates. The magnitude of this risk is evaluated as ρ1 , which is the norm of the first volatility component. By Proposition 6.5.1, we have the approximation ϕ1 ≈

R1 ρ1

(6.70)

under a low-volatility condition. This means that the first market price of risk ϕ1 is a risk-adjusted measure for the roll-down return of the entire forward rate curve. Thus, we have quantitatively obtained another interpretation of ϕ1 . The most important advantage of this observation is that the following systematic interpretation can be given for ϕi , i = 1, 2, 3, · · · . The market price of risk for investor We next examine the meaning of the second market price of risk in the principal component space by applying the above method. The bar graph in Fig. 6.8(b) represents a so-called steepening position, where the investor buys short-term bonds and sells long-term bonds. We may regard it as given that this portfolio’s value is not affected by the parallel shift of the entire forward rates (i.e., the first principal component of the covariance). Thus, the portfolio risk is hedged against the first volatility component. If the forward rate curve becomes steeper—for example, if the one-, two-, and three-year forward rates fall, and the four-, five-, and six-year forward

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rates rise, as shown by the dashed line in Fig. 6.8(b)—then the price of shortterm bonds will rise, but the price of the long-term bonds might fall2 . Then, the investor gains from both the buying and selling of bonds and expects the steepening trend implied by R2 < 0.

(6.71)

This investment style is known as the steepening strategy. The only risk in this strategy is from the flattening of the forward rate curve because the parallel shift risk is already hedged against. The magnitude of the slope change in this context is measured by the magnitude of the second volatility component ρ2 . Letting l = 2 in Corollary 6.5.1 gives ϕ2 = R2 /ρ2 ,

(6.72)

which is interpreted as the risk–return ratio of the steepening position. From this, the second market price of risk ϕ2 is a risk-adjusted measure for the steepening strategy. The converse portfolio strategy, in which short-term bonds are sold and long-term bonds are purchased, is called the flattening strategy. The risk– return ratio for this strategy is represented by ϕ2 . The third market price of risk and investment The bar graphs in Fig. 6.8(c) shows the so-called butterfly trading strategy or butterfly position, where the investor buys short- and long-term bonds and sells the middle-term bonds. The value of the butterfly position is affected by neither level changes nor slope changes for the same reason that the steepening position has these properties3 . In other words, the risk caused by the first and second volatility components is eliminated in this portfolio. The risk of the butterfly position is only that the curvature will change in the forward rate curve. 2

More precisely, for the argument on the second market price of risk, the prices of the four-year and five-year bonds might not fall when the forward rates become steeper. This is because the discount factors to four and five years are also affected by the falls of the shortterm forward rates. In practice, a steepening position is numerically designed to hedge against a risk of parallel shift. 3 The remarks in the above footnote apply to the butterfly position.

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If the curvature becomes large, for example, short- and long-term forward rates will fall, and middle-term forward rates will rise, as shown by the dashed line in Fig. 6.8(c). In that case, the prices of short- and long-term bonds rises, but the middle-term bonds fall in price. The investor gains from each bond and expects that the curvature becomes larger as R3 < 0. The narrow line shows the converse case. The magnitude of this risk is measured by the norm of the third volatility component ρ3 . Proposition 6.5.1 shows that, for the case l = 3, ϕ3 = R3 /ρ3 ; that is, the third market price of risk ϕ3 is a risk-adjusted measure for the butterfly trading strategy Accordingly, the higher-order market prices of risk can be interpreted in a similar manner. The market price of risk has been econometrically studied in relation to state variables such as the instantaneous spot rate r. For example, a functional √ form of ϕ(r) = a b2 + c2 r is assumed in De Jong and Santa-Clara (1999), where a, b, and c are constants. Stanton (1997) empirically estimates the market price of risk in the form of ϕ(r). In contrast, we provide a theoretical interpretation of the market price of risk in relation to the average changes in the historical forward rate curve, rather than to the state of the interest rate. 6.7

Property of Real-world Simulation

In the previous sections, we have systematically and qualitatively examined the numerical properties of the market price of risk. Next, we study the properties of the real-world model for simulation. In the following, interest rate simulation under the real-world measure is referred to as real-world simulation. Approach Before investigating the simulation properties, let us consider how to approach examining them. Monte Carlo simulation will yield too many numerical results, obscuring the features of the model. However, in cases where we use only a few random numbers for simulation, the results will depend on the specific

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numbers. Moreover, the results will depend on not only the volatility structure but also the initial forward rates. In light of this, such an experiment seems no better than ad hoc testing. Instead, it might be better to employ a theoretical approach for our objective. For this, we again work in the principal component space by using the exact formula (6.20) of the market price of risk. Real-world simulation We recall the relation T = x−t, which was given in Section 6.1. We set Ti = xi for t = 0 and i = 1, · · · , n. For a short time interval Δs > 0 and the initial forward rates f (0, Ti ), f (Δs, xi ) is represented under P by equation (6.2), as  Δs f (Δs, Ti ) = f (0, Ti ) + {−σ(s, Ti )υ(s, Ti ) + σ(s, Ti )ϕ}ds 0  Δs σ(s, Ti )dWs , (6.73) + 0

where Ws denotes a d-dimensional Brownian motion. We then have √ f (Δs, Ti ) = f (0, Ti ) + {−σ0i υ0i + σ0i ϕ}Δs + Δsσ0i W1 .

(6.74)

Here, we adopt the notation σ0i = σ(0, Ti ) and υ0i = υ(0, Ti ) for convenience. This equation presents a simulation model, substituting a d-dimensional number generated according to a standard normal distribution in W1 . l We define the volatility structure σ0i from the principal components, where the market price of risk is estimated by Theorem 6.2.1. Then, the single-period simulation is executed according to the above form. Real-world model in principal component space In particular, when d = n and (σ01 , · · · , σ0n ) is rank n, the above interest rate model is referred to as a full-factor model. In this section, we work with this model. The lth projection of equation (6.100) can be described as n  i=1

f (Δs, Ti )eli =

n 

f (0, Ti )eli +

i=1

n  {−σ0i υ0i eli + σ0i ϕeli }Δs i=1

n  √ + Δsσ0i eli W1 . i=1

(6.75)

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This is a representation of the real-world model in a principal component space.  We set βl = ni=1 σ0i υ0i eli , with l = 1, · · · , d. By the orthonormality of the set of principal components, it holds that n 

σ0i ϕeli

=

n  n 

l ρm e m i ϕm ei

i=1 m=1

i=1

= ρl ϕ l

(6.76)

and n 

σ0i eli W1

=

i=1

=

n n  

m l ρm e m i W1 e i

i=1 m=1 ρl W1l .

(6.77)

From these, we have n 

f (Δs, Ti )eli

=

i=1

n  i=1

f (0, Ti )eli + {−βl + ρl ϕl }Δs +



Δsρl W1l .

(6.78)

From Theorem 6.2.1, we already know that ϕl ρl = E H [αl ] + βl . Substituting this into the above, we obtain n  i=1

f (Δs, Ti )eli =

n  i=1

$ " #% f (0, Ti )eli + −βl + E H [αl ] + βl Δs

√ + Δsρl W1l n  √ = f (0, Ti )eli + E H [αl ] Δs + Δsρl W1l .

(6.79)

i=1

This represents a simulation model in the principal component space. Properties of simulation model Lemma 6.4.1 gives E H [αl ] = Rl . Then, equation (6.79) becomes n  i=1

f (Δs, Ti )eli =

n 

f (0, Ti )eli + Rl Δs +



Δsρl W1l

(6.80)

i=1

for l = 1, · · · , n. In practice, this structure is comprehensive. Indeed, the first term is the lth principal component score of the initial rates. The drift term is equal to the lth

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Takashi Yasuoka

rolled trend, and the diffusion coefficient is equal to the size of the lth volatility component. In other words, we may say that a real-world simulation model consists of a historical drift and a historical volatility in PCA representation. It is worth noting that this feature follows from the exact formulation of the market price of risk in Theorem 6.2.1. 6.8

Simulation Model in State Space

This section converts the real-world model in the principal component space to the real-world model in the state space. From the simulation having the form of equation (6.80) with l = 1, · · · , n, we have

⎞ ⎛ n ⎞ √ 1 1 1 f (0, T + R Δs + f (Δs, T )e )e W Δsρ 1 i i 1 i i 1 i=1 i=1 ⎜ ⎟ ⎜ ⎟ .. .. = ⎠ ⎝ ⎝ ⎠ . (6.81) . . √ n n n n n Δsρn W1 i=1 f (Δs, Ti )ei i=1 f (0, Ti )ei + Rn Δs + ⎛ n

Following the method in Appendix B, we multiply both sides of equation (6.81) by (e1 , · · · , en ) to represent the model in the state space. Then, the left-hand side becomes ⎛ n ⎞ 1 i=1 f (Δs, Ti )ei ⎜ ⎟ .. (e1 , · · · , en ) ⎝ ⎠ . n n i=1 f (Δs, Ti )ei ⎛ ⎞ f (Δs, T1 ) ⎜ ⎟ .. = (e1 , · · · , en )(e1 , · · · , en )T ⎝ ⎠ . f (Δs, Tn ) ⎛ ⎞ f (Δs, T1 ) ⎜ ⎟ .. =⎝ (6.82) ⎠. . f (Δs, Tn )

We calculate each term in the right-hand side of equation (6.81) as follows. The leftmost term becomes ⎛ n ⎞ ⎛ ⎞ 1 f (0, T ) f (0, T )e i 1 i i=1 ⎟ ⎜ ⎟ ⎜ .. .. (e1 , · · · , en ) ⎝ (6.83) ⎠=⎝ ⎠. . . n n f (0, Tn ) i=1 f (0, Ti )ei

Gaussian HJM Model

Interest Rate Modeling for Risk Management 143

From equation (6.58), the next term is reduced to ⎛ ⎞ ⎛ n ⎞ H 1 R1 Δs i=1 E [ΔFi /Δt]ei Δs ⎜ ⎟ ⎟ .. .. 1 n ⎜ (e1 , · · · , en ) ⎝ ⎠ = (e , · · · , e ) ⎝ ⎠ . . n H n E [ΔF /Δt]e Δs Rn Δs i i i=1 ⎞ ⎛ H E [ΔF1 /Δt]Δs ⎟ ⎜ .. (6.84) = ⎝ ⎠. . H E [ΔFn /Δt]Δs

l Since σ0i = ρl eli , the third term becomes ⎛ ⎞ ⎛ n ⎞ l l ρ1 W11 l=1 e1 ρl W1 ⎟ ⎜ ⎟ ⎜ .. .. (e1 , · · · , en ) ⎝ ⎠ ⎠ = ⎝ . .  n l l n ρn W 1 l=1 en ρl W1 ⎛ n ⎞ l l l=1 σ01 W1 ⎜ ⎟ .. = ⎝ ⎠. n . l l l=1 σ0n W1

(6.85)

Summarizing these, we have the simulation model represented in state space as   √ H ΔFi Δs + Δsσ0i W1 . (6.86) f (Δs, Ti ) = f (0, Ti ) + E Δt From this, we have obtained the following theorem. Theorem 6.8.1 For a full-factor Gaussian HJM model, let σ0i be given by l the principal components of the sample covariance as σ0i = ρl eli . Let ϕ be the market price of risk as obtained by Theorem 6.2.1. Then, for an arbitrary initial forward rate f (0, Ti ) and a short time step Δs with 0 < Δs < T1 , f (Δs, Ti ) is simulated by   √ H ΔFi Δs + Δsσ0i W1 ; i = 1, · · · , n. (6.87) f (Δs, Ti ) = f (0, Ti ) + E Δt From this theorem, the full-factor simulation is achieved, eliminating the market price of risk. Furthermore, the drift term is made equal to the rolled trend, and the volatility is made equal to the historical volatility. In this context, we may say that the real-world model represents a historical model

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admitting an arbitrage-free pricing system. Arbitrage pricing in the real-world model Let us trace the examination from Theorem 6.2.1 up to Theorem 6.8.1. First, we derived the formula for the market price of risk in Theorem 6.2.1. Next, we substituted this formula into the discretized forward rate process. Working with the full factor model, we eliminated the market price of risk in the real-world model, formulating it in the principal component space as equation (6.80). As a result, we have represented a simulation model in the state space in the above theorem, and this result is a historical term structure model. Some readers might question whether any historical term structure model is arbitrage-free. On this issue, let us recall the definition of an arbitrage-free market from Section 3.1, and that the arbitrage-free condition is ensured by the existence of a state price deflator. Then, the difference between the real-world model and the historical term structure model is that the former includes the market price of risk and the state price deflator, but the latter does not. Indeed, once we obtain the market price of risk ϕ, the state price deflator is implied by equation (4.26), such that dξt /ξt = −r(t)dt − ϕt dWt .

This is solved as    t   t |ϕ|2 ds − ϕdWs . −r(t) − ξt = exp 2 0 0 Using the Euler approximation, we obtain ξΔs by    √ |ϕ|2 Δs − sϕW1 . −f (0, 0) − ξΔs = exp 2

(6.88)

(6.89)

(6.90)

Consider a security with a payoff at T given by a random variable X. Then, the time t price pt of this security is given by equation (3.5) as pt = ξ1t Et [ξT X]. Thus, the arbitrage price is numerically computable in the real-world model. Key points for Theorem 6.8.1 There are two key items in deriving Theorem 6.8.1. The first is discretization by the Euler integral, which simplifies the discrete-time representation of

Gaussian HJM Model

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the forward rate evolution in equation (6.5). Indeed, if we attempt analytic estimation of υ(r, T ), we find that the approximation (6.5) is nontrivial in a continuous-time framework. The second is that the HJM framework admits an arbitrary volatility structure in a multi-dimensional model. Most studies on term structure modeling begin with a one-factor model and extend that model to a multi-factor model. In contrast, the previous section starts the investigation from a full-factor model, where PCA allows converting the model from the principal component space to the state space. Dimension reduction The full-factor model is not feasible for numerical simulation because of the large volatility matrix. Here, we investigate a method to reduce the dimensionality of the simulation model. For convenience, we recommend using equation (6.87) rather than equation (6.100) because the drift term −σ0i υ0i + σ0i ϕ in equation (6.100) should be computed for full factors. In contrast, the drift term E H [ΔFi /Δt] Δs in equation (6.87) is independent to the dimensionality. From this starting point, √  l we need only to approximate the diffusion term Δsσ0i W1 = nl=1 σ0i W1l by a d-factor model with d < n and use the form given in equation (6.87). Thus, we may determine the dimension d such that the accumulated contribution rate Cd is sufficiently close to 1. As a result, a d-dimensional model for real-world simulation is described as f (Δs, Ti ) = f (0, Ti ) + E

H



 d √  ΔFi l Δs + Δs σ0i W1l . Δt l=1

(6.91)

Remark In a strict sense, equation (6.91) is not supported in the arbitrage-pricing framework because the state price deflator ξt in equation (6.89) is valid for only the full-factor model. Of course, in practice, we may approximate this state price deflator ξt .

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6.9

Takashi Yasuoka

Numerical Procedure

To complete this chapter, we summarize a numerical procedure to build the real-world model in the Gaussian HJM framework. Let xi , i = 1, · · · , n be a sequence of time lengths, and set δi = xi+1 − xi . For a positive constant Δt > 0, we set the sequence of observation dates as tk = Δtk for k = 1, · · · , J + 1.

Numerical calculation of ΔFi (tk ) We practically observe the forward rates in the form F (tk , xi ) at tk for a fixed time length xi . Recall the definition of ΔFi in equation (6.12): ΔFi (tk ) = F (tk+1 , xi − Δt) − F (tk , xi ).

(6.92)

Usually, we set δi = 0.5 or 1 (years), and the time step Δt should be taken to be shorter than δ1 . Then, we may assume that Δt < mini δi . For this, the time length xi − Δt is not contained in the set of {xi }i=1,··· ,n . Thus, F (tk+1 , xi − Δt) is not directly observed in the market. The most convenient way to obtain F (tk+1 , xi − Δt) is to use F (tk+1 , xi ) as a proxy. However, the rolled trend exactly determines the drift of the simulation and the market price of risk, as mentioned in Section 6.5. This means that we should avoid using F (tk+1 , xi ) in place of F (tk+1 , xi − Δt). For the sake of expedience, F (tk+1 , xi − Δt) is approximately calculated by linear interpolation:   Δt Δt F (tk+1 , xi − Δt) = 1 − F (tk+1 , xi−1 ). (6.93) F (tk+1 , xi ) + δi−1 δi−1 Thus, we approximately obtain ΔFi (tk ). The importance of this interpolation will be verified in a numerical example, Case B4 in Section 10.2.3, which is given in the LMRW. Calculation of the market price of risk Below, we list the procedure to estimate the market price of risk from Theorem 6.3.2. For this, the rank of the covariance matrix is assumed to be n. 1. F (tk+1 , xi −Δt) is obtained by the interpolation (6.93), and then ΔFi (tk ) is calculated by equation (6.92) for i = 1, · · · , n, k = 1, · · · , J.

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Gaussian HJM Model

2. By PCA of Cov(ΔFi , ΔFj )/Δt, the eigenvalue (ρl )2 , the principal component el , and the volatility σ l are obtained for l = 1, · · · , n. 3. The number of factors d is chosen such that the accumulated contribution rate Cd is sufficiently close to 1. 4. The lth volatility component σ l is determined by l = ρl eli ; i = 1, · · · , n, l = 1, · · · , d. σ0i

5. The rolled trend is calculated as   H ΔFi E ; i = 1, · · · , n. Δt 6. From equation (4.9), υ0i is obtained by solving the integral  xi σ(0, u)du ; i = 1, · · · , n. υ0i = −

(6.94)

(6.95)

(6.96)

0

7. Next, σ0i υ0i is computed from σ0i υ0i =

d  l=1

l l υ0i ; i = 1, · · · , n. σ0i

(6.97)

More precisely, the right-hand side is an approximation of σ0i υ0i when d < n and is in agreement when d = n. 8. From Theorem 6.3.2, we calculate the lth market price of risk by solving    n  1  H ΔFi + σ0i υ0i eli ; l = 1, · · · , d. ϕl = E ρl i=1 Δt

(6.98)

Real-world simulation Lastly, we summarize the procedure for single-period simulation of forward rates by using equation (6.91). Here, the volatility is inferred by PCA and the market price of risk is eliminated. 1. The procedure from 1) to 7) is the same as that given above.

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Takashi Yasuoka

2. We set the initial instantaneous forward rate f (0, xi ) and generate a d-dimensional sequence of standard normal random numbers. 3. Identifying the time period xi with the maturity date, we set Ti = xi . Let Δs (< T1 ) be the time period for simulation. From equation (6.91), f (Δs, Ti ) is simulated by f (Δs, Ti ) = f (0, Ti ) + E

H



 d √  ΔFi l σ0i W1l . Δs + Δs Δt l=1

(6.99)

When we use a low dimensinal model, it is better to adopt the original drift in equation (6.100) instead of the second term in (6.99). Specifically, f (Δs, Ti ) is simulated by f (Δs, Ti ) = f (0, Ti ) + {−σ0i υ0i + σ0i ϕ}Δs +



Δsσ0i W1 .

Some numerical examples will be presented in Chapter 10.

(6.100)

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149

Chapter 7 REMARKS ON REAL-WORLD MODELS

Abstract: This chapter presents some remarks about real-world modeling. First, we examine the numerical differences between real-world simulations and riskneutral simulations, comparing the drift terms for the model types. Then, we investigate why the market price of risk is negative. This investigation is motivated by the following research question: Why does long-period observation tend to imply a negative value for the market price of risk? We introduce some simplified models (specifically, the flat yield model and the positive slope model) to answer this question. Next, we study the dynamical properties of the market price of risk. The mean price property of the market rice of risk in introduced to facilitate this. Throughout this book, we estimate the market price of risk under the assumption that it is constant during the sample period. Addressing this, we examine the validity of the constancy assumption for risk management by using a simplified model and the mean price property of the market price of risk. Additionally, Section 7.5 introduces the basic concepts of calculating credit exposure for counterparty credit risk management. Since some credit exposure should be calculated using the real-world probability, the real-world model benefits from this subject. It is worth nothing here that Section 7.1 closely follows Yasuoka (2015); and Sections 7.2, 7.3, and 7.4 are newly written for this book.

Keywords: Constant market price of risk, Counterparty credit risk, Economic scenario, ESG, Expected exposure, Flat yield model, Interest rate risk management, Long-period observation, Mean price property, Negative price tendency, Null market price of risk, Positive slope model, Potential future exposure, Real-world model, Risk management, Risk-neutral model, Timevarying market price of risk .

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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7.1

Takashi Yasuoka

Differences between Real-world and Risk-neutral Models

In this section, we investigate the differences in interest rate simulation between real-world models and risk-neutral models. Although there are multiple riskneutral models, characterized by the choice of a num´eraire, for this analysis, we employ the spot measure model as a benchmark for risk-neutral models. To highlight the differences, we compare the expectations from forward rate simulation under both measures, doing so for the reasons explained at the beginning of Section 6.7. We have the following theorem, which evaluates the expectations for each measure. Theorem 7.1.1 Under the assumptions of Theorem 6.8.1, f (Δs, Ti ) is normally distributed under both P and Q, and the expectations of f (Δs, Ti ) under each measures are given by   H ΔFi Δs, (7.1) E[f (Δs, Ti )] = f (0, Ti ) + E Δt E Q [f (Δs, Ti )] = f (0, Ti ) − σ0i υ0i Δs, (7.2) respectively. The variance of f (Δs, Ti ) in both simulations is Δs

n 

l 2 (σ0i ).

(7.3)

l=1

Proof Note that we are working with the full-factor model under the conditions of Theorem 6.8.1. From equation (6.87), f (Δs, Ti ) is represented by   √ H ΔFi (7.4) Δs + Δsσ0i W1 . f (Δs, Ti ) = f (0, Ti ) + E Δt distributed under P with mean n l 2 ). f (0, Ti )+E H [ΔFi /Δt] Δs and variance Δs l=1 (σ0i By the argument as in Section 4.2, when we assume the null market price of risk, the model is equivalent to the spot measure model. Then, substituting ϕ = 0 into equation (6.100) in Section 6.7, we have the discretization under Q as √ (7.5) f (Δs, Ti ) = f (0, Ti ) − σ0i υ0i Δs + Δs σ0i Z1 , Thus,

f (Δs, Ti )

is

normally

Remarks on Real-World Models

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where Z1 is an n-dimensional standard normal distribution under Q. Thus, f (Δs, Ti ) is normally distributed under Q with mean f (0, Ti ) − σ0i υ0i Δs and  l 2 variance Δs nl=1 (σ0i ) . This completes the proof. ✷ Δs

EQ[f(ΔS,Ti )] f(0,Ti ) Δs

0

T

Figure 7.1: Initial forward rate and the expectation of the simulation in the riskneutral model. E Q [f (∆s, Ti )] is approximately equal to the parallel shift of f (0, Ti ) to the left for ∆s when the volatility is low. Property of drift As described in the remark after Proposition 6.5.1, it sometimes happens that |σ0i υ0i |Δs takes a negligibly small value. In such cases, the equation (7.2) can be approximated by E Q [f (Δs, Ti )] ≈ f (0, Ti ). Fig. 7.1 illustrates this relation, where the bold line represents the initial forward rates f (0, Ti ), and the dashed line represents E Q [f (Δs, Ti )]. Specifically, the expectation E Q [f (Δs, Ti )] is almost equal to the parallel shift in the initial curve to the left for Δs. We usually observe this property when working with a term structure model for derivatives pricing. In contrast, equation (7.1) indicates that the drift of the real-world model is equal to the rolled trend. The following corollary roughly explains the difference between them. Corollary 7.1.2 Under the assumptions of Theorem 7.1.1, if it holds that   H ΔFi |σ0i υ0i | ≪ E , (7.6) Δt then the difference between the two model expectations is given by   Q H ΔFi Δs. E[f (Δs, Ti )] − E [f (Δs, Ti )] ≈ E Δt

(7.7)

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From Theorem 7.1.1, it follows that   Q H ΔFi E[f (Δs, Ti )] − E [f (Δs, Ti )] = E Δs + σ0i υ0i Δs. Δt

Takashi Yasuoka

Proof

(7.8)

From the assumption in equation (7.6), we may neglect the second term of the right-hand side to obtain an approximation. This gives equation (7.7), which completes the proof. ✷ The inequality (7.6) is the same as equation (6.60) in Proposition 6.5.1; this is satisfied under a large rolled trend with low volatility, as explained after the proof of Proposition 6.5.1. Naturally, this condition is not satisfied under a small rolled trend with high volatility. An example of this will be given in the LMRW (Section 10.2.1, Case A). Choice of sample period With respect to managing the risk of interest-rate rises, we sometimes hear the widely accepted explanation that the risk-neutral simulation implies valuation more on the safe side than the real-world simulation does. The reasoning is the following. The market price of risk might be negative, and the case of null market price of risk is equivalent to the spot measure model, so the real-world model will simulate lower interest rates than the spot measure model. However, it is possible that E H [ΔFi /Δt] ≫ 0. Succinctly, interest rates may rise rapidly. In this case, the real-world model will simulate higher interest rates than the risk-neutral model, which follows from the above corollary. In fact, an example of a positive market price risk will be given in Section 10.3.

Remarks on Real-World Models

7.2

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Negative Price Tendency of Market Price of Risk

In Section 4.2, we referred to the traditionally given reason for the negative market price of risk. Starting from there, we theoretically elucidate the mechanism for this, using the formula for the market price of risk. In empirical analysis of the term structure of interest rates, we often find that the observation period is long, such as thirty years. For example, De Jong (2000), Duffee (2002), and Cheridito et al. (2007) typically use thirtyor forty-year observation periods in analyzing the U.S. Treasury market to estimate the term structure and the market price of risk. When comparing the performance of short rate models, working with longer periods provides a better examination. However, for valuing a constant market price of risk, it is not clear that a long-period observation is advantageous. Indeed, it seems that long-period observation tends to imply a negative market price of risk. This section examines that tendency. To simplify the argument for clarity and convenience, we introduce some simple term structures: a flat yield model and a positive slope model. 7.2.1

Flat Yield Model

Let xi , i = 1, · · · , n be a sequence of time lengths with xi+1 − xi = δ > 0, and let F (t, xi ), i = 1, · · · , n denote the forward rates observed on t ∈ [0, ∞). We assume that the yield is always flat. Then, the forward rates F (t, xi ), i = 1, · · · , n are also flat, and so we may denote them by F (t, xi ) = F (t). We call this interest rate model a flat yield model. This model represents sample data, but it is not assumed to be an Ito process. Let [0, τ ] be a sample period with 0 < τ < ∞. From PCA for the sample, we obtain an eigenvalue (ρ1 )2 , the principal component e1 , and the first volatility component σ 1 . To distinguish according to the length of the observation period ([0, τ ]), we denote these values by ρ(τ ), e(τ ) and σ(τ ), respectively, and omit the superscript “1” for the order in principal components. Let us estimate the market price of risk in the flat yield model, applying Theorem 6.2.1. By the flat yield assumption, the first principal component is

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also flat. From this, the condition 1 ei (τ ) = √ ; i = 1, · · · , n. n

n

i=1 (ei (τ ))

2

Takashi Yasuoka

= 1 implies that (7.9)

The volatility σ0i (τ ) is expressed as σ0i (τ ) = ρ(τ )ei (τ ) ρ(τ ) = √ , n

(7.10)

and we denote this by setting σ(τ ) = σ0i (τ ) for simplicity. From equation (4.10), υ0i (τ ) is obtained from υ0i (τ ) = −σ(τ )δi. The constant β(τ ) is defined with respect to τ by equation (6.16). Taking these together with equation (6.16), β(τ ) is represented by n n  (σ(τ ))2 δ  σ(τ )υ0i (τ )ei = − √ β(τ ) = i n i=1 i=1 √ 1 = − (σ(τ ))2 δ n(n − 1). (7.11) 2 Because of the flatness assumption and from equation (6.55), the rolled trend coincides with the observable trend. Then, from equations (6.53) and (6.57), the first rolled trend score R(τ ) is given by R(τ ) = O(τ ) = √

n  F (τ ) − F (0) √ nτ i=1

n(F (τ ) − F (0)) . (7.12) τ We denote by ϕ(τ ) the first market price of risk implied from the sample period [0, τ ]. Recall that Lemma 6.4.1 presents = E H [α]. From this, ϕ(τ ) is estimated by Theorem 6.2.1 as =

ϕ(τ ) = 7.2.2

R(τ ) + β(τ ) . ρ(τ )

(7.13)

Negative Price Tendency

By using equation (7.13), let us examine a sufficient condition for ϕ(τ ) < 0. The sign of ϕ(τ ) coincides with the sign of R(τ ) + β(τ ) in equation (7.13)

Remarks on Real-World Models

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because we always set ρ(τ ) > 0. Combining equation (7.11) with (7.12), we have √ √ n(F (τ ) − F (0)) 1 R(τ ) + β(τ ) = − (σ(τ ))2 δ n(n − 1). (7.14) τ 2 Then, the following theorem holds, which provides a sufficient condition for ϕ(τ ) < 0 in the flat yield model. Theorem 7.2.1 (Negative price condition) ever F (τ ) − F (0) δ(n − 1) < (σ(τ ))2 , τ 2

In a flat yield model, when-

(7.15a)

it holds that ϕ(τ ) < 0. Conversely, if F (τ ) − F (0) δ(n − 1) > (σ(τ ))2 , τ 2 then ϕ(τ ) > 0.

(7.15b)

The next example shows that the inequality (7.15a) is satisfied when the observation period is sufficiently long. Example 7.2.1 We set δ = 0.5 (years), n = 20, and τ = 20 (years). Suppose that the volatility is 5%; that is, σ(20) = 0.05. We assume that the forward rates are always positive and less than 20%; specifically, 0 < F (t) < 0.2 for t ∈ [0, 20]. These assumptions should be sufficiently natural for our experience. In this case, the left-hand side of equation (7.15a) is evaluated as 0.2 − 0 F (20) − F (0) < = 0.01. (7.16) τ 20 This means that the yield rises by less than 1% per year, on average. The right-hand side of equation (7.15a) is calculated as 19 × 0.5 × (0.05)2 = 0.011875, (7.17) 2 which is larger than the left-hand side. These figures satisfy the inequality (7.15a), and so the market price of risk is estimated to be negative.

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Specifically, the market price of risk ϕ(20) is estimated as follows. The value of ρ(20) is calculated from equation (7.10) as ρ(20) =



20σ(20) = 4.47 × 0.05 = 0.2236.

(7.18)

From these figures and equation (7.13), ϕ(20) is calculated to be −0.0084 as follows: ϕ(20)
0 be fixed and tk = Δtk, k = 1, · · · , J + 1 be a sequence of observation dates. For an observation dataset F (tk , xi ), we set ΔFi (tk ) = F (tk+1 , xi − Δt) − F (tk , xi ).

(8.12)

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From the covariance of ΔFi (tk ), PCA identifies the principal components el and the eigenvalues ρl for l = 1, · · · , n. We approximate the first volatility component ρ1 e1i by σ exp(−κxi ), with σ and κ chosen such that ρ1 e1i ≈ σ exp(−κxi ).

(8.13)

Consider the least squares problem n  i=1

{ρ1 e1i − σ exp(−κxi )}2

(8.14)

under the restriction 2

(ρ1 ) = σ

2

n 

exp(−2κxi ).

(8.15)

i=1

Solving this yields σ and κ. The condition (8.15) is referred to as the norm-invariant condition; it ensures that the implied Hull–White volatility has the same norm as that of the first volatility component. If we do not impose this condition, then the implied Hull–White volatility may have a norm different from that of the first volatility component ρ1 . This difference makes the numerical procedure unnecessarily complicated for real-world modeling. To avoid this, we assume the norm-invariant condition, which will be successfully applied for the estimation of the market price of risk in Section 8.4 Once we have determined σ and κ, we obtain the Hull–White volatility by setting σ(0, xi ) = σ exp(−κxi ) for each i. Humped volatility The volatility structure σ(0, xi ) is not always observed to be exponentially decaying on i. Sometimes, we observe a humped volatility structure; that is, the volatility increases from the spot rate to the middle-term forward rate and decreases in the long term. Using an actual example, we give some remarks about the calibration.

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1st VC Case A Case B Case C

0.010 0.008 0.006 0.004 0.002

i

0.000 1

3

5

7

9

11

13

15

17

19

Figure 8.2: Approximation of the first volatility component by the Hull–White volatility “1st VC” represents the first volatility component. Implied forward rates are obtained from the Japanese LIBOR swap market from 16 June 2008 to 31 August 2009. Interest rate data are the same as in Fig. 8.1. The forward rates and the principal volatility component were calculated by the author.

Example 8.3.1 (Humped volatility) Set δ = 0.5 (year) and i = 1, · · · , 20. From the same data as in Example 8.2, we obtain the implied six-month forward rates, which we regard as the instantaneous forward rates. The bold line in Fig. 8.2 shows the first volatility component, which is humped—increasing up to i = 9 and decreasing from i = 9 to i = 20. This is troublesome when modeling the Hull–White volatility σ exp(−κx) because this function is monotone on x. The second column in Table 8.2 shows the first principal component of the historical data, as derived by PCA. Example 8.3.2 (Calibration for humped volatility) To fit the Hull–White volatility to the humped structure, we consider three calibration styles, as follows. In Case A, we estimate the parameters σ and κ from the principal component for i = 1, · · · , 20, which corresponds to entire evolution of the forward rates. In Case B, we estimate the parameters with i = 1, · · · , 10, which corresponds to the dynamics of the short- and middleterm forward rates. In Case C, we estimate the parameters with i = 10, · · · , 20, which corresponds to the evolution of the middle- and long-term forward rates.

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Table 8.1: Volatility fitting in the Hull–White model. “1st VC” represents the first volatility component. Forward rates and the volatility component are the same as in Fig. 8.2.

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1st VC 0.00264 0.00481 0.00558 0.00620 0.00676 0.00720 0.00753 0.00774 0.00779 0.00767 0.00744 0.00715 0.00675 0.00623 0.00566 0.00512 0.00471 0.00449 0.00433 0.00421

Case A i = 1, · · · , 20 0.00656 0.00651 0.00647 0.00643 0.00639 0.00635 0.00630 0.00626 0.00622 0.00618 0.00614 0.00610 0.00606 0.00602 0.00598 0.00595 0.00591 0.00587 0.00583 0.00579

σ exp(−κxi ) Case B Case C i = 1, · · · , 10 i = 10, · · · , 20 0.00461 0.00495 0.00531 0.00569 0.00611 0.00655 0.00703 0.00754 0.00809 0.00868 0.00795 0.00743 0.00695 0.00649 0.00607 0.00567 0.00530 0.00496 0.00463 0.00433 0.00405

The results are shown in Fig. 8.2 and Table 8.1. The third column in Table 8.1 and the narrow solid line in Fig. 8.2 show the Hull–White volatility estimated in Case A, with σ and κ shown in Table 8.2, where the mean reversion rate is seen to be slightly positive, at 0.0130. Thus, the volatility is modeled with weakly positive mean reversion for the entire forward rates. In Case B, the Hull–White volatility is shown as the fourth column in Table 8.1 and the bold dashed line in Fig. 8.2. From Table 8.2, the mean reversion rate is seen to be remarkably negative, at −0.1405; that is, the volatility is modeled so as to represent a negative mean reversion for short- and middleterm forward rates.

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The rightmost column of Table 8.1 and the narrow dashed line in Fig. 8.2 show the Hull–White volatility in Case C. From Table 8.2, we see that the mean reversion rate is obviously positive, at 0.1350. Thus, the volatility is modeled so as to represent a positive mean reversion for middle- and long-term forward rates. This humped structure is typically observed, rather than being a rare event, and so we reasonably hesitate over which method to choose. Although we work with the Hull–White model here, which method for calibration is reasonable depends on the properties of the interest rate risk specific to each financial institution and each portfolio. When a financial institution (or portfolio) has mostly short-term interest rate risk, the method of Case B is appropriate. However, when a financial institution has both short- and long-term interest rate risk, the method of Case A may be appropriate. Making this choice is a financial engineering matter. For comparison, the rightmost column in Table 8.2 exhibits the parameter obtained in Section 8.2, that is, the volatility implied by the short rate distribution. This result is quite different from all those of Cases A, B, and C. Humped volatility model Furthermore, if we intend to model the humped volatility structure in its entirety, it might be better to depart from the Hull–White model. The humped volatility is modeled in Bhar et al. (2002), Mercurio and Moraleda (2000), and Moraleda and Vorst (1997), among others, in the form σ(t, T ) = σ{γ(T − t) + 1} exp{−κ(T − t)},

(8.16)

where σ, γ, and κ are positive constants. This function is humped when γ > κ. For details, readers should consult Brigo and Mercurio (2007) and Nawalkha Table 8.2: Parameters in the Hull–White model for three cases. The column labeled “Short rate” represents the results from Section 8.2.

σ κ

Case A 0.0066 0.0130

Case B 0.0043 -0.1405

Case C 0.0156 0.1350

Short rate 0.0318 0.0049

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et al. (2007). It might be possible to construct the real-world model with this volatility as an application of the argument in the next section. However, when working with non parametric models, there is a way to work with the first principal component as the volatility in the HJM model such that σ(0, xi ) = ρ1 e1i for i = 1, · · · , n, where (ρ1 )2 is the first eigenvalue and e1 is the first principal component; that is, we would be considering a one-factor HJM model. For that case, the real-world model can be built as shown in Section 6.9. 8.4

Real-world Modeling

In this section, we present a method of calculating the market price of risk in a one-factor Hull–White model, applying Theorem 6.3.2. The approach here is basically taken with reference to real-world modeling in the Gaussian HJM model. First, we note an important difference between the Hull–White model and the Gaussian HJM model in real-world modeling. For this, we recall the simulation form (6.87) in a full-factor Gaussian HJM model, which is f (Δs, Ti ) = f (0, Ti ) + E

H



 √ ΔFi Δs + Δsσ0i W1 . Δt

(8.17)

Here, the second term is determined from the observed rolled trend for each i, and we can use an arbitrary volatility structure for σ0i . In the Hull–White model, the volatility is parametrically given by the function σ exp{−κ(T − t)}, and the drift term is determined from this volatility and the market price of risk. From this, the drift term is different from the observed rolled trend due to the use of one-dimensional modeling. First volatility component Let tk , k = 1, · · · , J + 1 and xi , i = 1, · · · , n be the same as in Section 8.3. Suppose that we have already obtained the first eigenvalue (ρ)2 and the first principal component (e1 , · · · , en )T . For clarity, we denote ρ1 as ρ in this section and do the same with other variables. To determine the parameters κ and σ, recall the least squares problem

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(8.14), n  i=1

{ρei − σ exp(−κxi )}2 .

(8.18)

We determine σ and κ by finding the solution that minimizes the above equation and satisfies the norm-invariant condition (8.15), n  2 2 exp(−2κxi ). (8.19) (ρ) = σ i=1

From these parameters, we set the volatility σ0i from the Hull–White volatility such that σ0i = σ exp(−κxi ) ; i = 1, · · · , n.

(8.20)

We may also call (σ01 , · · · , σ0n )T the Hull–White volatility. Next, we define an n-dimensional vector (˜ e1 , · · · , e˜n )T by e˜i =

σ exp(−κxi ) ; i = 1, · · · , n. ρ

(8.21)

From equation (8.19), we see that n 

(˜ ei )2 = 1.

(8.22)

i=1

Hence, we may regard (˜ e1 , · · · , e˜n )T as the first principal component. Since it holds that ei ; i = 1, · · · , n, (8.23) σ0i = ρ˜ the Hull–White volatility (σ01 , · · · , σ0n )T can be regarded as the first volatility component. Thus, the volatility structure σ0i is determined by equation (8.20) or, equivalently, by equation (8.23). Norm-invariant condition A remark on the norm-invariant condition, which we imposed on the least squares problem (8.18), might help here. In numerical analysis, the norm of the minimizing solution might be slightly different from that of the original

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first volatility component, (ρe1 , · · · , ρen )T . For this reason, equation (8.22) does not necessary hold, and so we impose the condition (8.19) on the approximation. This means that we obtain the Hull-White volatility as a new volatility component, and this setup makes it easy to calculate the market price of risk, which we will see later. Calculation of β By equation (4.49), we have σ0i υ0i = σ(0, xi )υ(0, xi ) σ2 (8.24) = − exp(−κxi ){1 − exp(−κxi )} ; i = 1, · · · , n. κ From this relation, we obtain the constant β, defined in equation (6.16) by β=

n 

σ0i υ0i e˜i .

(8.25)

i=1

Drift term It is not trivial to apply Theorem 6.3.2 to the Hull–White model because of the use of one-dimensional modeling with parametric volatility. So we must carefully develop our argument, which follows. To estimate the market price of risk within the Gaussian HJM framework, we recall αl (tk ), defined in equation (6.15) as n

1  ΔFi (tk )eli ; l = 1, · · · , d, k = 1, · · · , J. αl (tk ) = Δt i=1

(8.26)

We note that this equation is valid for a multi-factor model where the principal volatility components are employed and the rank of the covariance is equal to d. If the observed data ΔFi (tk ) follow the one-factor Hull–White model, then we may assume that this is the case in which d = 1. In reality, market data do not typically satisfy this assumption. Therefore, we artificially determine α(tk ) as α(tk ) =

n  ΔFi (tk ) i=1

Δt

e˜i ; k = 1, · · · , J.

(8.27)

Hull–White Model

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Fortunately, we can regard e˜ as the first principal component, and so Proposition B.2.1 implies that the above α(tk ) is the solution to the approximation ΔFi (tk ) ≈ α(tk )˜ ei . Δt

(8.28)

This means that the change in the forward rates (ΔF1 (tk ), · · · , ΔFn (tk ))T are approximated by a simple scaling of the Hull-White volatility. In fact, from the following argument, we shall see that we do not need to compute α(tk ) for each tk . Market price of risk Note that we are working in the one-dimensional principal component space determined by the Hull–White volatility. Theorem 6.3.2 implies that    n  1 H ΔFi ϕ= + σ0i υ0i e˜i . E (8.29) ρ i=1 Δt

Substituting equations (8.25) and (8.27) into the above, we obtain an alternative expression of ϕ as ϕ = ρ1 (E H [α] + β). From the definition (6.58), we calculate the first rolled trend score R as   n  H ΔFi R= E e˜i . (8.30) Δt i=1 Lemma 6.4.1 asserts that E H [α] = R. From this, we have ϕ=

R+β . ρ

(8.31)

This means that we can obtain ϕ without needing α(tk ), k = 1, · · · , J, which is the reason that we do not need to calculate them (as stated above). We note that there might be a more elegant method to derive the form (8.31). 8.5

Simulation Model

We finally present a form for simulation under P. Set Ti = xi for i = 1, · · · , n; here, Ti indicates the date and xi indicates the time length. Let f (0, Ti ), i = 1, · · · , n be an initial forward rate.

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We remark that we cannot use the form (6.87), f (Δs, Ti ) = f (0, Ti ) + E

H



 √ ΔFi Δs + Δsσ0i W1 , Δt

(8.32)

because this is valid for only the full-factor model. From the argument for the drift term in the previous section, it is practical to work with the form (6.100) for simulation in the Hull–White model, as √ f (Δs, Ti ) = f (0, Ti ) + {−σ0i υ0i + σ0i ϕ}Δs + Δsσ0i W1 . (8.33) Substituting equations (8.20) and (8.24) into the above, we obtain   2 √ σ −κxi −κxi f (Δs, Ti ) = f (0, Ti ) + e (1 − e ) + σ0i ϕ Δs + Δsσ0i W1 κ  σ (1 − e−κxi ) + ϕ Δs = f (0, Ti ) + σe−κxi κ √ −κxi + Δsσe W1 . (8.34) This is the form of the real-world simulation in the Hull–White model. From this and equation (8.31), we obtain the following theorem. Theorem 8.5.1 (1) In a one-factor Hull–White model, the market price of risk is estimated by ϕ = (R + β)/ρ, where R and β are obtained by equations (8.30) and (8.25), respectively, and (ρ)2 is the first eigenvalue. (2) A single-period simulation for a short time period Δs is given by σ  f (Δs, Ti ) = f (0, Ti ) + σe−κxi (1 − e−κxi ) + ϕ Δs κ √ −κxi W1 . (8.35) + Δsσe Similarly to Theorem 7.1.1, the following corollary specifies the distributions of f (Δs, Ti ) under P and Q, for which the proof is obvious. Corollary 8.5.2 In a one-factor Hull–White model, let us consider two simulation forms for f (Δs, Ti ): by equation (8.35) under P, and by equation (8.11) under Q.

Hull–White Model

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(1) f (Δs, Ti ) is normally distributed under P and Q, and the expectations under each measures are given by σ  −κxi −κxi E[f (Δs, Ti )] = f (0, Ti ) + σe (1 − e ) + ϕ Δs, (8.36) κ σ2 (8.37) E Q [f (Δs, Ti )] = f (0, Ti ) + e−κxi {1 − e−κxi }Δs. κ The variance of f (Δs, Ti ) in both models is equal to Δsσ 2 exp(−2κxi ). (2) The expected difference in f (Δs, Ti ) between the real-world model and the risk-neutral model is given by E[f (Δs, Ti )] − E Q [f (Δs, Ti )] = σe−κxi ϕΔs.

(8.38)

From the second result, the real-world model simulates higher (resp., lower) forward rates than the risk-neutral model under Q when the market price of risk is positive (resp., negative). We remark that the dimensional reduction to a one-factor model cannot avoid introducing model risk when Cd is not sufficiently close to 1. This is a typical problem in financial engineering for one-dimensional modeling. Nelson–Siegel model Although this book does not study real-world modeling by using the Nelson– Siegel term structure, this term structure is sometimes employed in the study of economics. Here, we briefly address ourselves to the real-world modeling of the Nelson–Siegel term structure model. By a method analogous to the argument in Section 8.4, it is possible to construct a two- or three-factor model using the Nelson–Siegel function. Let Ti and xi be the same as above, and let a two-dimensional volatility be given by the Nelson–Siegel function as σ 1 (0, T ) = 1

(8.39a)

σ 2 (0, T ) = e−c1 T − c2 ,

(8.39b)

where c1 and c2 are positive constants; σ 1 (0, T ) represents a parallel shift in the forward rate curve, which corresponds to the first principal component; and σ 2 (0, T ) represents a slope change in the curve, which corresponds to the

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second principal component. In particular, c1 corresponds somewhat to the mean reversion rate in the Hull–White model. Next, for some integers i and j with 1 ≤ i < j ≤ n, we determine c2 by

1 (8.40) c2 = (e−c1 xi + e−c1 xj ). 2 Here, i and j are chosen such that the volatility structure (8.39b) adequately explains the historical term structure. For example, we may set i = 1 and j = n. From this, it follows that 1 1 σ 1 = (σ01 , σ0n ) = (1, 1)

(8.41a)

2 2 σ 2 = (σ01 , σ0n ) = (e−c1 x1 − c2 , e−c1 xn − c2 ).

(8.41b)

A direct calculation from equation (8.40) implies that 1 2 1 2 σ01 + σ0n σ0n σ 1 σ 2 = σ01

= (e−c1 x1 − c2 ) + (e−c1 xn − c2 ) = 0.

(8.42)

Then, σ 1 is orthogonal to σ 2 when we consider only two grid points, at x1 and xn . We may regard σ 1 and σ 2 as volatility components obtained by PCA. Combining this with the argument in Section 8.4 for the Hull–White model, the market price of risk is estimated in the manner developed in Section 6.2. The rest is entrusted to readers. 8.6

Numerical Procedure

We already see that the Hull–White model is simple to construct as a realworld model. This section summarizes a numerical procedure for estimating the market price of risk and constructs a Monte Carlo simulation model. Since the calculation is completed in a one-dimensional model, the subscript “1” for vectors to indicate that they are the first factor number is omitted. Numerical examples will be shown in Sections 10.3 and 10.4. Let xi , tk , and ΔFi (tk ) be the same as in Section 6.9; here, i = 1, · · · , n and k = 1, · · · , J + 1.

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Market price of risk 1) F (tk+1 , xi − Δt) is obtained by the interpolation (6.93), and ΔFi (tk ) is given by equation (6.92) for i, k, i = 1, · · · , n, k = 1, · · · , J. 2) The eigenvalue (ρ)2 and the principal component e = (e1 , · · · , en )T are obtained by PCA for the covariance Cov(ΔFi (tk ), ΔFj (tk ))/Δt. Here, we set ρ such that ρ > 0. 3) We determine the parameters σ and κ by solving the least squares problem (8.14), n  i=1

{ρei − σ exp(−κxi )}2 ,

under the norm-invariant condition, (ρ)2 =

(8.43) n

i=1

σ 2 exp(−2κxi ).

4) We define the n-dimensional vector (˜ e1 , · · · , e˜n ) by σ e˜i = exp(−κxi ) ; i = 1, · · · , n, ρ

(8.44)

and set the Hull–White volatility from σ0i = ρ˜ ei for i = 1, · · · , n. 5) From equations (8.24) and (8.25), we calculate σ0i υ0i as σ2 (8.45) exp(−κxi ){1 − exp(−κxi )}. κ 6) The constant β is calculated from equations (8.44) and (8.45) as β = n ˜i . i=1 σ0i υ0i e σ0i υ0i = −

7) We calculate the rolled trend E H [ΔFi /Δt] from ΔFi (tk ) for all i. 8) By equation (8.30), the first rolled trend score is obtained by   n  H ΔFi R= e˜i . E Δt i=1

(8.46)

9) The market price of risk is estimated from equation (8.31) as ϕ = (R + β)/ρ.

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Monte Carlo simulation 1) Set xi = Ti for i = 1, · · · , n. 2) Let f (0, Ti ), i = 1, · · · , n be an initial forward rate. 3) Determine the volatility structure and estimate the market price of risk according to the previous procedures, from 1) to 9). 4) From equation (8.35), a single-period simulation for a short time interval Δs under P is represented by −κxi



f (Δs, Ti ) = f (0, Ti ) + σe (1 − e κ √ + Δsσe−κxi W1 ,

−κxi



) + ϕ Δs (8.47)

with i = 1, · · · , n. Here, W1 is a one-dimensional standard normal distribution.

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1

Chapter 9 REAL-WORLD MODEL IN THE LIBOR MARKET MODEL

Abstract: This section of the book develops the theory of simulation in the LMRW. Although the theory for this is developed similarly to that for the Gaussian HJM model, the results here are somewhat more complicated than those. In particular, the drift term in the LMRW has an additional feature that makes it different from that in the Gaussian model. Moreover, the methods for reducing dimensionality and constructing the drift term for use in simulation are different from those for the Gaussian model. Readers are recommended to review the corresponding results in Chapter 6 to more deeply understand the properties of the LMRW. Most of the arguments in this chapter are based on Yasuoka (2013a); Section 9.2 is newly written to describe maximum likelihood estimation for the market price of risk. Some numerical examples will be shown in Chapter 10, using the same historical data as in the example for the Gaussian HJM model.

Keyword: Complete market, Dimensionality reduction, Eigenvalue, Full-factor model, Interpretation of the market price of risk, Least squares problem, LIBOR market model, LMRW, Log-scale observable trend, Log-scale rolled trend, Market price of risk, Monte Carlo simulation, Maximum likelihood estimation, MPR score, Negative price tendency, PCA, Principal component score, Properties of simulation, Real-world measure, Real-world simulation, State space setup, Volatility component .

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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We have already studied a real-world model, the Gaussian HJM model, in Chapter 6. The drawback of the Gaussian model is that it generates negative interest rates. To allow satisfying the practical requirement of a positive interest rate model, this section develops a theory of simulation in the LMRW. The structure of the arguments used to develop this model are parallel to those applied to the Gaussian HJM model, following Yasuoka (2013a). 9.1

Discretization of LIBOR Process

In this section, we introduce a basic setup to describe the observation of LIBOR data in the framework of the LMRW. Setup of LIBOR processes Let τ > 0 be a time horizon, and let 0 < T1 < · · · < Tn+1 = τ be a sequence of maturity dates such that Ti+1 − Ti = δi > 0 for all i. Then, xi is defined as the time length to Ti at t = 0 for all i. In this chapter, we assume that the market is complete and arbitrage-free, meaning that the market price of risk is determined uniquely. We denote by Li (t) the forward LIBOR observed at t over the period [Ti , Ti+1 ], and by λi (t) a d-dimensional deterministic volatility of Li (t), where we assume that 0 < d ≤ n. A d-dimensional process χi (t) is defined in equation (5.9) as χi (t) =

λi (t)δi Li (t) . 1 + δi Li (t)

(9.1)

We remark that we sometimes use Li , λi , and χi instead of Li (t), λi (t), and χi (t), respectively, for the sake of brevity. Other variables are treated the same. In the LMRW, the LIBOR process is expressed from equation (5.10) as ⎧ ⎛ ⎫ ⎞  t i ⎨ t ⎬ 2  |λi | ⎠ ⎝λi χj + λ i ϕ − Li (t) = Li (0) exp λi dWt , (9.2) dt + ⎩ 0 ⎭ 2 0 j=m(t)

for i = 1, · · · , n, where Wt is a d-dimensional Brownian motion under P. Let {tk }k=1,··· ,J+1 be a sequence of observation times with tk+1 − tk = Δt > 0, where J + 1 (J > 0) is the number of observation times and Δt > 0 is a

Libor Market Model

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K(tk, xi ) K(tk+1, xi -Δt)

K(tk+1, xi )

tk tk+1 tk+2

K(tk+2, xi -Δt) T-t=x xi-Δt

xi

Time to maturity

Figure 9.1: Observation of forward LIBOR. The forward LIBOR is observed in the form K(t, xi ), for a fixed value of xi , at tk , tk+1 , and tk+2 . constant. The Euler approximation of equation (9.2) implies ⎫ ⎧ i ⎨  2⎬ |λi | χj (t) + λi ϕ − Δt log Li (t + Δt) = log Li (t) + λi ⎩ 2 ⎭ j=m(t) √ + Δtλi W1 , 1 where W1 = 0 dW is a d-dimensional standard normal distribution.

(9.3)

Observation of forward LIBOR For simplicity, we assume in the following that Ti+1 − Ti = δ > 0 for all i with T1 = δ. Setting T − t = x and xi = δi for all i, we denote by K(tk , xi ) the implied forward LIBOR on the period [tk + xi , tk + xi+1 ] as observed at time tk . Further, we assume that K(tk , xi ) is continuously extended to K(t, x) for arbitrary t and x. Then, it holds that Li (t) = K(t, xi − t) ; i = 1, · · · , n.

(9.4)

We usually obtain the forward LIBOR from the LIBOR/swap rates in the form K(t, δi), rather than in the form Li (t), because we observe the term structure of the LIBOR/swap rates with respect to the term length xi . Fig. 9.1 visually illustrates this observation. The dashed lines exhibit forward LIBOR curves

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observed at times tk , tk+1 , and tk+2 , and the forward LIBOR is written in the form K(t1 , xi ), · · · , K(tJ+1 , xi ) for a fixed xi . We assume that the volatility corresponding to the process K(tk − t, δi − t) with t ∈ [0, Δt] is given by λi (0) for all k. Additionally, we write λ0i = λi (0) for convenience. With this, we define a sequence1 κj (tk ), k = 1, · · · , J + 1 as κj (tk ) =

λ0j δK(tk , δj) ; k = 1, · · · , J + 1. 1 + δK(tk , δj)

(9.5)

Replacing χj (t) of equation (9.1) with κj (tk ), equation (9.3) becomes log K(tk + Δt, xi − Δt) = log K(tk , xi ) +



λ0i

|λ0i |2 +λ0i ϕ − 2



Δt +

i 

κj (tk )

j=1



Δtλ0i W1 ,

(9.6)

where we set m(tk ) = 1 in equation (9.3) for all k with tk = 0 + 0. For this, we remark that if tk = 0 then it originally holds that m(t) = 0. However, equation (9.6) is obtained from integration of t ∈ (0, Δt], where m(t) = 1. Then, we  may substitute m(tk ) = 1 into the term ij=m(tk ) κj (tk ) in equation (9.6). 9.2

Estimation of the Market Price of Risk

This section presents the maximum likelihood estimation of the market price of risk in the LIBOR market model. PCA In the manner developed in Section 6.2, we represent the LIBOR process in a principal component space. A sample covariance matrix V is obtained from Vij =

1

1 Cov (log K(tk + Δt, xi − Δt) − log K(tk , xi ), Δt log K(tk + Δt, xj − Δt) − log K(tk , xj )) ,

(9.7)

We note that the symbol κj here does not indicate the mean reversion rate from the Hull–White model.

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for i, j = 1, · · · , n. We assume that the covariance matrix V has rank d ≤ n. Similarly to the approach in Section 6.2, V is decomposed into Vij =

d  l=1

eli (ρl )2 elj ; 1 ≤ i ≤ n, 1 ≤ j ≤ n,

(9.8)

where (ρl )2 is the lth eigenvalue, and el = (el1 , · · · , eln ) is the lth principal component. Here, we always assume that el1 > 0, ρl > 0 ; l = 1, · · · , d,

(9.9)

and that all eigenvalues are distinct. This assumption is the same as that used in the real-world model for the Gaussian HJM model. Since V is a symmetric matrix, the set of principal components e1 , · · · , ed is orthonormal by Proposition A.2.2; specifically, n  i=1

eli ehi = δlh ; 1 ≤ l, ≤ d, 1 ≤ h ≤ d.

(9.10)

Principal component space We set the principal volatility component as λl0i = ρl eli ; i = 1, · · · , n, l = 1, · · · , d.

(9.11)

Next, we represent the dynamics of the forward LIBOR by using PCA. Similarly to that approach in Section 6.2, we call λl = (λl01 , · · · , λl0n )T the lth principal volatility component, or the lth volatility component for short. It follows that l 2

|λ |

=

n 

(ρl eli )2

i=1

= (ρl )2 .

= |ρl |

2

n 

(eli )2

i=1

(9.12)

From this, ρl indicates the magnitude of the lth volatility component λl . Since the rank of V is d, the n × d matrix (λ1 , · · · , λd ) has rank d. We set ˜ i (yk ) as ΔK ˜ i (tk ) = log K(tk+1 , xi − Δt) − log K(tk , xi ). ΔK

(9.13)

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With this, equation (9.6) becomes   i 2  √ |λ | 0i ˜ i (tk ) = κj (tk ) + λ0i ϕ − ΔK λ0i Δt + Δtλ0i W1 2 j=1   i d 2   |λ | 0i Δt = λ0i κj (tk ) + λm 0i ϕm − 2 m=1 j=1 √

+ Δt

d 

m=1

m λm ; i = 1, · · · , n. 0i W1

(9.14)

Consider a d-dimensional principal component space spanned by e1 , · · · , ed . Regarding ! T ˜ 1 (tk ) ˜ n (tk ) ΔK ΔK (9.15) ,··· , Δt Δt ˜ l (tk ), k = 1, · · · , J as as an n-dimensional vector in Rn , we define a sequence α the lth projection of the vector by setting α ˜ l (tk ) =

n  ˜ i (tk ) ΔK

Δt

i=1

eli ; l = 1, · · · , d.

(9.16)

Similarly, we define a d-dimensional sequence β˜l (tk ) as β˜l (tk ) =

n  i=1



i 

|λ0i |2 λ0i κj (tk ) − 2 j=1



eli ; l = 1, · · · , d.

(9.17)

We remark that the constant vector (β1 , · · · , βd )T was defined in the Gaussian HJM model by equation (6.15). In contrast, κj (tk ) in the right-hand side of equation (9.17) depends on tk , and β˜l (tk ), too, is defined as depending on tk . Dividing equation (9.14) by Δt, and projecting the result into el , we have, from (9.16), that α ˜ l (tk ) = β˜l (tk ) +

n  d 

i=1 m=1

n

l λm 0i ϕm ei

d

1  m m l λ0i W1 ei . +√ Δt i=1 m=1

(9.18)

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From equation (9.11) and the orthonormality expressed in equation (9.10), we have α ˜ l (tk ) = β˜l (tk ) +

d 

m=1

ρm ϕ m

n  i=1

l em i ei

d n  1  l m +√ em ρm W 1 i ei Δt m=1 i=1

1 (9.19) = (β˜l (tk ) + ρl ϕl ) + √ ρl W1l . Δt This is an expression of equation (9.14) in the principal component space. This form differs from the corresponding equation (6.17) in the Gaussian HJM model as follows: • α ˜ l (tk ) is measured in a logarithmic scale; and • β˜l (tk ), k = 1, · · · , J is a sequence in tk , and involves the term −|λ0i |2 /2. Maximum likelihood estimation The last term in the right-hand side of equation (9.19) is normally distributed. From Proposition C.0.1, the maximum likelihood estimation of the market price of risk ϕl is obtained as the solution to the least squares problem of θl (ϕl ): J  θl (ϕl ) = |˜ αl (tk ) − {β˜l (tk ) + ρl ϕl }|2 ; l = 1, · · · , d. (9.20) k=1

Note that this formulation is basically the same as the least squares problem (6.19) in the Gaussian HJM framework, and each θl (ϕl ) is a one-variable function of ϕl . We can solve d minimizing problems for each θl (ϕl ), l = 1, · · · , d, independently, to obtain ϕ = (ϕ1 , · · · , ϕd )T . We have the following theorem, which is similar to Theorem 6.2.1. Theorem 9.2.1 For the LIBOR market model, we assume that the volatility corresponding to K(tk + t, δi − t) with t ∈ [0, Δt] is given by the principal volatility component such that λl0i is defined by equation (9.11). Then, the lth market price of risk, ϕl , is uniquely given by  1  H E [˜ αl ] − E H [β˜l ] ; l = 1, · · · , d, (9.21) ϕl = ρl

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where α ˜ l (tk ) =

n  ˜ i (tk ) ΔK

Δt

i=1

β˜l (tk ) =

n  i=1

Proof



eli ; l = 1, · · · , d,

i 

|λ0i |2 λ0i κj (tk ) − 2 j=1



(9.22)

eli ; l = 1, · · · , d.

(9.23)

By direct calculation,

J  ∂θ(ϕl ) = −2 [α ˜ l (tk ) − {β˜l (tk ) + ρl ϕl }]ρl , ∂ϕl k=1

(9.24)

and ∂ 2 θ/∂ϕ2l = 2(ρl )2 J > 0 for all l = 1, · · · , d. Since θ(ϕl ) is a one-variable function of ϕl and ρl = 0, θ(ϕl ) is strictly convex with respect to ϕl . The least squares solution then exists uniquely, and can be written as the root of the equation J

 ∂θ(ϕl ) = −2ρl {α ˜ l (tk ) − (β˜l (tk ) + ρl ϕl )} = 0. ∂ϕl k=1

(9.25)

Since ρl = 0, the above equation is reduced to 0 =

J  k=1

{α ˜ l (tk ) − (β˜l (tk ) + ρl ϕl )}

= ρl ϕl J +

J  k=1

{α ˜ l (tk ) − β˜l (tk )}.

(9.26)

 The sample means of α ˜ l (tk ) and β˜l (tk ) are represented by E H [α ˜ l ] = J1 Jk=1 α ˜ l (tk ) J ˜ 1 H ˜ and E [βl ] = J k=1 βl (tk ), respectively. Substituting these into equation (9.26), the term ρl ϕl is represented as

ρl ϕ l

J 1 = {α ˜ l (tk ) − β˜l (tk )} J k=1

= E H [α ˜ l ] − E H [β˜l ].

Then, the lth market price of risk ϕl is formulated as ϕl = This completes the proof.

(9.27) 1 {E H ρl

[α ˜ l ]−E H [β˜l ]}. ✷

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From this formulation of ϕl , we can develop a theory of simulation in the LMRW analogous to that developed in the Gaussian HJM model. Comparison with the Gaussian HJM model We here give a remark on the expression of the market price of risk in the LMRW in comparison with that in the Gaussian HJM model. In the LMRW, the market price of risk is formulated as equation (9.21), in which we can see that the sign for the term E H [β˜l ] is negative. In contrast, the market price of risk for the Gaussian HJM model is formulated, in equation (6.20), as ϕl = ρ1l {E H [αl ] + βl }, in which we see the sign for the constant term βl is positive, and so opposite that in the LMRW. To verify this difference, we recall the forward rate process (4.17) in the HJM model as df (t, T ) = {−σ(t, T )υ(t, T ) + σ(t, T )ϕ(t)} dt + σ(t, T )dWt .

(9.28)

Here, the process υ(t, T ) was defined by equation (4.10), which includes a negative sign, as  T υ(s, T ) = − σ(s, u)du. (9.29) s

 From this, the constant βl is defined, by equation (6.16), as βl = ni=1 σ0i υ0i eli , and so the sign of βl is opposite that of the HJM volatility σ(t, T ). In the LMRW, the LIBOR process is given, in (5.10), as ⎫ ⎧ i ⎬ ⎨  dLi (t) = λi (t) χj (t) + λi (t)ϕt dt + λi (t)dWt . (9.30) ⎭ ⎩ Li (t) j=m(t)

With this, the observation of the LIBOR is represented, in (9.6), as  i  log K(tk + Δt, xi − Δt) = log K(tk , xi ) + λ0i κj (tk ) j=1

+λ0i ϕ −

|λ0i | 2

2



Δt +



Δtλ0i W1 ,

(9.31)

where κj (tk ) =

λ0j δK(tk , δj) . 1 + δK(tk , δj)

(9.32)

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˜ k ) is defined, by equation (9.17), as Taking this as given, the sequence β(t   i n 2   |λ | 0i eli . (9.33) λ0i κj (tk ) − β˜l (tk ) = 2 i=1 j=1 Then, the sign of β˜l (tk ) coincides with that of the LIBOR volatility, λ. Along these lines, the sign of E H [β˜l ] in equation (9.21) is opposite to that of βl in equation (6.20). In this sense, the formulation (9.21) in the LMRW is consistent with the formulation (6.20) in the Gaussian HJM model. 9.3

State Space Setup

This section estimates the market price of risk in a state space setup, corresponding to the argument in the Gaussian HJM framework in Section 6.3. Here, we denote the market price of risk by ϕ′ = (ϕ′1 , · · · , ϕ′d )T to avoid confusion with the results from the previous section. We suppose the same conditions as in Section 9.2, where the principal ˜ i (tk ) from equation (9.13) volatility components are employed. Substituting ΔK into equation (9.6), we have   i 2  √ |λ | 0i ′ ˜ i (tk ) = λ0i κj (tk ) + λ0i ϕ − Δt + Δtλ0i W1 , (9.34) ΔK 2 j=1 for all k and i. We denote by ǫ(ϕ′ ) the sum of the squared differences between both sides of the above equation in the time series and cross sections, neglecting the random part, such that !  n J  i   1 ˜ i (tk ) − λ0i ΔK κj (tk ) ǫ(ϕ′ ) = J k=1 i=1 j=1  2 |λ0i |2 ′ Δt . (9.35) +λ0i ϕ − 2 Let ϕ′ be the solution that minimizes ǫ(ϕ′ ). According to Chapter 6, we call this setting the state space setup, and we call the setting in the previous section the PCA setup. The differences between ϕ′ and ϕ (as defined in the previous section) are as follows.

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• ϕ is the solution that minimizes θ(ϕ) from equation (9.20) in the PCA setup, and is the maximum likelihood estimate of the market price of risk. • ϕ′ is the solution that minimizes ǫ(ϕ′ ) of equation (9.35) in the state space setup. As it was in the study of the Gaussian HJM model, the derivation of ǫ(ϕ′ ) is immediate; further, it is more comprehensive than that of θ(ϕ), because ǫ(ϕ′ ) is defined without the use of PCA. Accordingly, ϕ′ is defined more directly than the PCA setup. However, it is not trivial in this setup to obtain ϕ′ as the maximum likelihood estimate because the structure of equation (9.34), needed for maximum likelihood estimation, is quite complicated. With respect to the minimizing problem of ǫ(ϕ′ ), we obtain the following theorem, which is analogous to Theorem 6.3.2. Theorem 9.3.1 The solution to the least squares problem for ǫ(ϕ′ ) is uniquely given by  & ' & i '  n  ˜ |λ0i |2 1  H Δ Ki H ′ E − λ0i E eli , κj + (9.36) ϕl = ρl i=1 Δt 2 j=1 for l = 1, · · · , d.

Proof

From the definition of the sample mean, it holds that

J 1 ˜ ˜ i] ΔKi (tk ) = E H [ΔK J k=1

(9.37)

and & i ' i J  1  H κj (tk ) = E κj . J k=1 j=1 j=1 Substituting these into equation (9.35), we have  !  & i ' n 2   |λ | 0i ˜ i ] − λ0i E H Δt ǫ(ϕ′ ) = E H [ΔK κj + λ0i ϕ′ − 2 i=1 j=1

(9.38)

2

. (9.39)

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To avoid confusion, the subscripts i and l express the length of time xi to the maturity Ti and the ordering index of the Brownian motion Wl , respectively.  From λ0i ϕ′ = dl=1 λl0i ϕ′l , it follows that ∂λ0i ϕ′ = λl0i ; l = 1, · · · , d, i = 1, · · · , n. ∂ϕ′l

With this, the partial derivative of ǫ(ϕ′ ) in ϕ′l is given by ! & i '  n   ∂ǫ(ϕ′ ) ˜ i ] − λ0i E H E H [ΔK = 2 κj ∂ϕ′l i=1 j=1   |λ0i |2 ′ +λ0i ϕ − Δt λl0i Δt ; l = 1, · · · , d. 2

(9.40)

(9.41)

Also, it holds that

n

 ∂ 2 ǫ(ϕ′ ) 2 λl0i λm ; l, m = 1, · · · , d. = 2 0i Δt ′ ∂ϕl ∂ϕ′m i=1

(9.42)

λl0i = ρl eli ; l = 1, · · · , d, i = 1, · · · , n.

(9.43)

We recall that the lth volatility component is given, by equation (9.11), as

From the orthonormality in equation (9.10), we have 2

n 

2 2 λl0i λm 0i Δt = 2ρl ρm δlm Δt .

(9.44)

i=1

Thus, the Hessian matrix of ǫ(ϕ′ ) is diagonal because ∂ 2 ǫ(ϕ′ ) = 2(Δtρl )2 δlm . ′ ′ ∂ϕl ∂ϕm

(9.45)

The assumption (9.9) ensures that the Hessian is positive definite. Then, ǫ(ϕ′ ) is strictly convex with respect to ϕ′ . Hence, the solution ϕ′ = (ϕ′1 , · · · , ϕ′d )T to the least squares problem is uniquely determined by solving ∇ǫ(ϕ′ ) = 0. For simplicity of expression, we define a constant vector γ = (γ1 , · · · , γn )T as & & i ' '  ˜ |λ0i |2 H Δ Ki H γi = E κj + − λ0i E ; i = 1, · · · , n. (9.46) Δt 2 j=1

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Note that γ is completely determined from the historical data. Substituting equation (9.46) into equation (9.41), the equation ∇ǫ(ϕ′ ) = 0 can be decomposed to n  i=1

(γi − λ0i ϕ′ )λl0i = 0 ; l = 1, · · · , d.

(9.47)

By equation (9.43), we have the following. n 

λ0i ϕ′ λl0i =

i=1

n 

λl0i

=

ρl eli

d 

d 

ϕ′m ρm em i

m=1

i=1

= ρl

ϕ′m λm 0i

m=1

i=1

n 

d 

ϕ′m ρm

m=1

n 

eli em i

(9.48)

i=1

The principal components el and em are orthonormal to each other for all l and m, and so the above equation can be reduced to n 

λ0i ϕ′ λl0i

= ρl

i=1

=

d 

ϕ′h ρh δlm

m=1 (ρl )2 ϕ′l

; l = 1, · · · , d.

(9.49)

γi eli ; l = 1, · · · , d.

(9.50)

Additionally, it holds that n  i=1

γi λl0i

=

n 

γi ρl eli

i=1

= ρl

n  i=1

Substituting these into equation (9.47), we have ρl ϕ′l = we obtain n 1  ′ γi eli ; l = 1, · · · , d. ϕl = ρl i=1

n

i=1

Substituting equation (9.46) into γi in the above, we have '   & & i ' n 2   ˜i Δ K |λ | 1 0i − λ0i E H eli . EH κj + ϕ′l = ρl i=1 Δt 2 j=1

γi eli . From this,

(9.51)

(9.52)

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This completes the proof. We have the following corollary, as an analogue of Corollary 6.3.3.



Corollary 9.3.2 Under the same conditions as in Theorem 9.3.1, the solution for ϕ′ that minimizes ǫ(ϕ′ ) coincides with the maximum likelihood estimate of the market price of risk ϕ; that is, ϕ′l = ϕl for l = 1, · · · , d. αl ] − Proof By equation (9.21) in Theorem 9.2.1, ϕ is given by ϕl = ρ1l {E H [˜ H ˜ E [βl ]} for each l. Substituting equations (9.16) and (9.17) into this, we have that & &  ' ' i n 2  ˜i |λ 1  | Δ K 0i κj (tk ) − ϕl = EH − E H λ0i eli ρl i=1 Δt 2 j=1  & & i ' '  n 2  ˜i Δ K |λ 1  | 0i EH = − λ0i E H eli , (9.53) κj + ρl i=1 Δt 2 j=1 for l = 1, · · · , d. The right-hand side is equal to ϕ′l in Theorem 9.3.1. This completes the proof. ✷ 9.4

Historical Trends of LIBOR

This section examines the numerical meaning of the market price of risk in connection with the average change in the forward LIBOR. For this, we make some definitions to describe the average change of the forward LIBOR in a manner similar to that introduced in Section 9.4. Log-scale observable trend Let tk , k = 1, · · · , J + 1, xi , i = 1, · · · , n and K(tk , xi ) be as defined in Section 9.1. The log-scale observable trend of forward LIBOR K( ·, xi ) is defined as E

H



 K(tk+1 , xi ) 1 log . Δt K(tk , xi )

It holds that   K(tk+1 , xi ) 1 K(tJ+1 , xi ) 1 H log = log . E Δt K(tk , xi ) ΔtJ K(t1 , xi )

(9.54)

(9.55)

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Then, the log-scale observable trend for xi is positive when K(tJ+1 , xi ) − K(t1 , xi ) > 0. Conversely, this trend is negative when K(tJ+1 , xi ) − K(t1 , xi ) < 0. The log-scale observable trend represents an average change in the log-scale forward LIBOR for a fixed xj during the sample period [t1 , tJ+1 ]. Precisely speaking, this trend indicates the averaged speed of log-scale change of the forward LIBOR. The fundamental property of this trend is basically the same as that explained in Fig. 6.2 for the HJM framework. Log-scale rolled trend ˜ i (tk ) as We set ΔK ˜ i (tk ) = log K(tk+1 , xi − Δt) − log K(tk , xi ), ΔK

(9.56)

˜ i /Δt]. Immediately, and define the log-scale rolled trend of K( ·, xi ) as E H [ΔK it holds that ' &   ˜i K(tk+1 , xi − Δt) 1 H Δ K H = E log . (9.57) E Δt Δt K(tk , xi ) Then, we have & ' ˜i 1 Δ K = − E H [log K(tk , xi ) − log K(tk+1 , xi − Δt)] EH Δt Δt   1 K(tk+1 , xi ) H +E log . Δt K(tk , xi )

(9.58)

The first term nearly represents the sample mean of the slope of the forward LIBOR curve in log-scale, and the second one is equal to the log-scale observable trend. Similarly to the rolled trend in the Gaussian HJM model, the log-scale rolled trend corresponds to the roll-down or roll-up of the forward LIBOR in log-scale. In the LMRW, we always work with the trends and the scores in log-scale, and so we will sometimes omit the term “log-scale” for convenience in the following. The next definition is analogous to Definition 6.4 in the Gaussian HJM framework.

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Definition 9.4 The lth principal component score of the observable trend ˜ l is defined as O   n  1 K(t , x ) k+1 i H ˜l = O E log eli ; l = 1, · · · , d. (9.59) Δt K(t , x ) k i i=1 ˜ l is defined Similarly, the lth principal component score of the rolled trend R as ' & n  ˜i Δ K ˜l = eli ; l = 1, · · · , d. (9.60) EH R Δt i=1

˜ l as the lth observable trend score and the lth rolled trend ˜ l and R We refer to O score, respectively, for short. ˜1, O ˜ 2 , and According to the argument in Sections 4.3 and 6.4, we see that O ˜ 3 respectively represent the level, slope, and curvature factors of the observO ˜1, R ˜ 2 , and R ˜ 3 respectively represent the level, slope, able trend. Similarly, R and curvature factors of the rolled trend. ˜2 ˜ 1 and R Examples for R ˜ 1 is explained similarly to the explanation in Section 6.4. From The value of R equation (9.58), it roughly holds that ˜1 ≈ O ˜ 1 − {Mean of the slope of forward LIBOR curve}. R

(9.61)

˜ 1 with respect to the From this, Table 9.1 qualitatively estimates the value of R slope of the forward LIBOR curve and the observable trend. Naturally, this table exhibits the same characteristics that Table 6.1 does. ˜ 1 with respect to the slope of the forward Table 9.1: Qualitative estimate of R LIBOR curve and the log-scale observable trend.

Slope of forward LIBOR curve Upward sloping Flat Downward sloping

Log-scale observable trend Fast fall Stable Fast rise Negative Negative Positive Negative Near zero Positive Negative Positive Positive

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The second rolled trend, too, is explained in the manner given in Section ˜ 1 and R ˜ 2 with respect to the 6.4. Table 9.2 roughly estimates the values R observable trend. Furthermore, the rolled trend scores of higher orders are explained in the same manner. 9.5

Qualitative Estimate of Market Price of Risk

In this section, we investigate the meaning of the market price of risk with respect to a term structure of the rolled trend. The next proposition is an analogue of Proposition 6.5.1. Proposition 9.5.1 Under the assumptions of Theorem 9.2.1, if it holds that  & i '     |λ |2    H ΔKi    0i H   (9.62) κj  + ≪ E λ0i E    2 Δt j=1 ˜ l /ρl for for all i, then the market price of risk ϕl is approximated by ϕl ≈ R every l = 1, · · · , d. Proof

From Theorem 9.3.1, the lth market price of risk is represented by '   & & i ' n 2   ˜ |λ0i | 1 Δ Ki − λ0i E H eli . ϕl = EH κj + (9.63) ρl i=1 Δt 2 j=1

By the assumption (9.62), we can approximate this as & ' n ˜i 1  H ΔK E eli . ϕl ≈ ρl i=1 Δt

(9.64)

˜ 1 and R ˜ 2 with respect to log-scale observable trends. Forward LIBOR Table 9.2: R curves are assumed to be upward-sloping during the sample period.

Observable trend Bull steep Steep Bear steep Bull flat Flat Bear flat

˜1 R Negative Negative Near zero Negative Negative Near zero

˜2 R Negative Negative Negative Positive Positive Positive

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Table 9.3: Qualitative estimate for the first market price of risk ϕ1 with respect to the observable trend.

Slope of forward LIBOR curve Upward-sloping Flat Downward-sloping

Observable trend Fast fall Stable Fast rise Negative Negative Positive Negative Near zero Positive Negative Positive Positive

˜ l /ρl . This comSubstituting equation (9.60) into the above, we obtain ϕl ≈ R pletes the proof. ✷ In Section 10.2, we will show an actual example that satisfies the inequality (9.62). However, the market data do not always satisfy this inequality. For example, let us suppose an interest rate market where the LIBOR volatility is lower than 0.2 (20%) when taking the long-range view. Then, we may assume that |λi | < 0.2 for all i. Recall that χj was replaced by κj in Section 9.1. From  equation (5.12), ij=1 κj indicates the volatility of the ith bond price, which would be absolutely lower than the LIBOR volatility. Then, we may assume  i  that  j=1 κj  = 0.01. With these, the left-hand side in equation (9.62) is estimated by  ' & i   |λ |2  0.22   0i ≪ 0.2 × 0.01 + κj (tk )  + λ0i E H   2 2 j=1 = 0.002 + 0.02

= 0.022 (2.2%).

(9.65)

However, the right-hand side of equation (9.62) represents an average speed of the rolled trend. It is not usual for the forward LIBOR to change more than 2% in a year. Therefore, the assumption (9.62) requires that the volatility is low and the forward LIBOR changes by a large amount. Although this condition is not always satisfied, Proposition 9.5.1 is useful to qualitatively explain the meaning of the market price of risk.

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Table 9.4: Qualitative estimates for ϕ1 and ϕ2 . Forward LIBOR curves are assumed to be upward-sloping in the sample period.

Observable trend Bull steep Steep Bear steep Bull flat Flat Bear flat

ϕ1 Negative Negative Near zero Negative Negative Near zero

ϕ2 Negative Negative Negative Positive Positive Positive

Rolled trend and the market price of risk From equation (9.12), ρl indicates the magnitude of the lth volatility component λl . As stated in Section 6.5, we may say that ρl indicates the lth volatility risk. With this, Proposition 9.5.1 states that the lth market price of risk ϕl means, roughly, the rolled trend score corresponding to the lth volatility risk; that is, {Magnitude of lth rolled trend} {lth market price of risk} ≈ . (9.66) {lth volatility risk}

˜ l , the above estimate is roughly dependent on the ob˜l ≈ R Assuming that O ˜ 1 with ϕ1 in Table 9.1, we obtain Table 9.3, which servable trend. Replacing R shows a qualitative estimate of the first market price of risk ϕ1 . In the manner described in Section 6.5, we obtain Table 9.4. This table shows rough estimates of ϕ1 and ϕ2 with respect to the observable trend, where the forward LIBOR curves are assumed to be upward-sloping in the sample period, and the condition (9.62) is assumed to hold. The market price of risk and investment Similarly to the argument in Section 6.6, we infer the meaning of the market price of risk in connection with a bond investment strategy. Here, the deduction follows the pattern of deduction in the Gaussian HJM model, we list only the results. For details, see the observation in Section 6.6. • The first market price of risk ϕ1 is a risk-adjusted measure for the rolldown return of the entire forward LIBOR curve.

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• The second market price of risk ϕ2 is a risk-adjusted measure for the steepening strategy. • The third market price of risk ϕ3 is a risk-adjusted measure for the butterfly trading strategy. Negative price tendency When the inequality (9.62) fails to hold, the market price of risk is not necessarily well-approximated by Proposition 9.5.1. In such a case, there is a possibility that the second term in equation (9.36) will be larger than the first term (the rolled trend). In this respect, long-period observation would indicate a negative price tendency for the first market price of risk in the LIBOR market model, as in Theorem 7.2.2. Remark The observations in this section were presented in Yasuoka (2013a) earlier than the analogous observation in the Gaussian HJM framework, which was given in Yasuoka (2015). If the market price of risk was calculated by statistical software, then we could obtain a numerical value only and might not consider these issues. Along these lines, we may say that quantitative estimation of the market price of risk makes studying real-world modeling theoretically feasible. Because of this, we continue to develop the theoretical foundation for real-world simulation in the following sections. 9.6

Fundamental Properties of Simulation

This section investigates the properties of simulation in the LMRW. Let the time interval Δs > 0 be fixed, and let λi (t) and γi (t) be constant for t ∈ [0, Δs]. Let Li (0) be an initial forward LIBOR. From equation (9.2), we have ⎧ ⎛ ⎞ i ⎨ Δs 2  |λi (t)| ⎠ ⎝λi (t) Li (Δs) = Li (0) exp χj (t) + λi (t)ϕ − dt ⎩ 0 2 j=m(t)   Δs λi (t)dWt ; i = 1, · · · n, (9.67) + 0

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where Wt is a d-dimensional Brownian motion under P. We assume a fullfactor model—specifically, that d = n and the n×n matrix λi has rank n—and examine the properties of real-world simulation. By the Euler approximation, Li (Δs) can be expressed as

log Li (Δs) = log Li (0) +

⎧ ⎨

λ0i





i 

j=m(t)

χ0j + λ0i ϕ −

+ Δsλ0i W1 ; i = 1, · · · , n.



2⎬

|λ0i | Δs 2 ⎭

(9.68)

Here, we let λ0i = λi (0) and χ0i = χi (0). Then, the single-period simulation for Li (Δs) is executed with this form. Projection to principal component space We assume that the principal volatility components are used for the volatility, such that λl0i is defined by equation (9.11). Then, the market price of risk is given by Theorem 9.2.1. In the manner introduced in Section 6.7, we examine the properties of equation (9.68) in the principal component space. The lth projection of equation (9.68) is represented as n  i=1

log Li (Δs)eli

=

n 

log Li (0)eli

i=1

+

⎧ n ⎨  i=1

+

n 

⎩ √

λ0i

i 

j=m(t)

χ0j eli + λ0i ϕeli −

Δsλ0i W1 eli .

2

⎫ ⎬

|λ0i | l e Δs 2 i⎭

(9.69)

i=1

We set constants bl , l = 1, · · · , d by letting   i n   |λ0i |2 bl = λ0i χ0j − eli . 2 i=1 j=1

(9.70)

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With this, equation (9.69) becomes n 

log Li (Δs)eli

=

n 

log Li (0)eli

i=1

i=1

+

n 



+

!

bl +

n 

λ0i ϕeli Δs

i=1

Δsλ0i W1 eli .

(9.71)

i=1

By the definition of λ0i and the orthonormality of the principal components, we have n  n n   l l ρm e m λ0i ϕei = i ϕ m ei i=1 m=1

i=1

= ρl ϕl ; l = 1, · · · , n

(9.72)

and n 

λ0i W1 eli =

i=1

=

n  n 

i=1 m=1 ρl W1l ;

m l ρm em i W1 e i

l = 1, · · · , n.

(9.73)

Substituting these into equation (9.71), we obtain n  i=1

log Li (Δs)eli

=

n 

log Li (0)eli + (bl + ρl ϕl )Δs +

√ Δsρl W1l .

(9.74)

i=1

From Theorem 9.2.1, we already have that ρl ϕl = E H [˜ αl ]−E H [β˜l ]. Here, α ˜ l (tk ) ˜ and βl (tk ) are given by n  ˜ i (tk ) ΔK (9.75) eli , α ˜ l (tk ) = Δt i=1   n i 2   |λ | 0i β˜l (tk ) = λ0i eli . (9.76) κj (tk ) − 2 i=1 j=1 From these, equation (9.74) becomes n n .   l H H ˜ l log Li (0)ei + bl + E [˜ αl ] − E [βl ] Δs log Li (Δs)ei = i=1

i=1

√ + Δsρl W1l ; l = 1, · · · , n.

(9.77)

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This represents a simulation model in the principal component space. Real-world simulation model Further. we modify the above equation to examine the properties of the drift term. From equations (9.70) and (9.76), it follows that  i n   |λ0i |2 H ˜ χ0j − λ0i bl − E [βl ] = 2 j=1 i=1 & i '   |λ0i |2 H κj + eli − λ0i E 2 j=1  & i ' n i    λ0i eli = χ0j − λ0i E H κj i=1

=

n 

j=1

λ0i

i=1



i  j=1

j=1

χ0j − E

H

&

i 

κj

j=1

'

eli .

(9.78)

Substituting this into equation (9.77), we have n  i=1

log Li (Δs)eli

=

n  i=1

+

" log Li (0)eli + E H [˜ αl ]

n  i=1

+



λ0i

!

i  j=1

Δsρl W1l

χ0j − E H

&

i  j=1

; l = 1, · · · , d.

κj

'

eli



Δs (9.79)

 αl ] and E H [ ij=1 κj ] are calculated from the sample data, and Note that E H [˜ the values of log Li (0), λ0j , and χ0j are determined from the initial data. Along these lines, equation (9.79) practically specifies the real-world simulation model in the principal component space.

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Remark on β˜l (tk ) and bl Here, we give a supplementary explanation about the subtle difference between two similar variables, β˜l (tk ) and bl . Initially, χi (t) is defined by equation (5.9) for introducing the LIBOR process in Section 5.3 as χi (t) =

λi (t)δi Li (t) ; i = 1, · · · , n. 1 + δj Li (t)

(9.80)

Here, we set the constant χ0i by letting λ0j δi Li (0) χ0i = χ( 0) = (9.81) 1 + δj Li (0) for the initial LIBOR Li (0). Then, the constant bl is determined by equation (9.70) with the initial LIBOR and the given volatility, resulting in   n i   |λ0i |2 bl = eli ; l = 1, · · · , d. (9.82) λ0i χ0j − 2 i=1 j=1 The sequences κj (tk ) and β˜l (tk ) are calculated by equations (9.5) and (9.76), respectively, for all sampled days tk as λ0j δK(tk , δj) κj (tk ) = ; j = 1, · · · , n, k = 1, · · · , J (9.83) 1 + δK(tk , δj) and   n i 2   |λ | 0i eli ; λ0i κj (tk ) − β˜l (tk ) = 2 i=1 j=1 l = 1, · · · , d, k = 1, · · · , J.

(9.84)

Hence, bl is a constant, but β˜l (tk ) is defined as a sequence in tk .

Approximation of simulation model In the LMRW, the properties of the simulation model are not simple to explain in the way that the properties of the Gaussian HJM model are. Keeping this in mind, we begin by working with a case chosen for the simplicity of examining it: We take the case where the initial LIBOR is equal to the historical mean. Specifically, Lj (0) = E H [K( ·, δj)].

(9.85)

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This assumption may seem somewhat general for studying the properties of the simulation. Indeed, we can find this as an initial condition in Geiger (2011, pp. 52–53). For this case, we expect that the following approximation holds:   λ0j δLj 0) λ0j δK( ·, δj) H ≈E ; j = 1, · · · , n. (9.86) 1 + δLj (0) 1 + δK( ·, δj) From this and equation (9.1), we have i i   λ0j δLj (0) χ0j = 1 + δLj (0) j=1 j=1 ' & i  λ0j δK( ·, δj) ; i = 1, · · · , n. ≈ EH 1 + δK( ·, δj) j=1

Substituting equation (9.5) into the above, we obtain & i ' i   χ0j ≈ E H κj ; i = 1, · · · , n. j=1

(9.87)

(9.88)

j=1

From this, the simulation having the form given in equation (9.79) is approximately reduced to n n   √ l log Li (Δs)ei ≈ (9.89) αl ] Δs + Δsρl W1l . log Li (0)eli + E H [˜ i=1

i=1

˜ l , by equations (9.75) and (9.60), respecRecalling the definitions of α ˜ l and R tively, we have & & ' ' n n  ˜ i (tk )el ˜ Δ K Δ K i i i=1 αl ] = E H = eli EH E H [˜ Δt Δt i=1

˜ l ; l = 1, · · · , n. = R (9.90) This relation is an analogue to the result from Lemma 6.4.1 in the Gaussian HJM framework. Substituting this into (9.89), we obtain n n   √ l ˜ l Δs + Δsρl W l . log Li (Δs)ei ≈ (9.91) log Li (0)eli + R 1 i=1

i=1

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Properties of simulation model The above approximation (9.91) admits the same structure as the real-world model (6.80) in the Gaussian HJM model: n 

f (Δs, Ti )eli

=

n 

f (0, Ti )eli + Rl Δs +



Δsρl W1l .

(9.92)

i=1

i=1

Both models are characterized by the lth rolled trend score for the drift term and historical volatility for the diffusion term. The difference between them is that the form (9.91) is an approximation represented in log-scale, while the initial condition is set by equation (9.85). On the basis of this observation, we may say that the simulation model approximately consists of historical drift and historical volatility in the principal component space representation, while the initial LIBOR is given by the sample mean as equation (9.85). To make a more general argument, in the next section, we will examine the properties of the drift term under an arbitrary initial condition. 9.7

Real-world Model in State Space

In the previous section, we studied the properties of the simulation in the principal component space, where the initial forward LIBOR is assumed to satisfy equation (9.85). This section expands on this by allowing an arbitrary initial condition. From equation (9.79), Li (Δs) is represented in the principal component space by n  i=1

log Li (Δs)eli

=

n  i=1

+

" log Li (0)eli + E H [˜ αl ]

n  i=1

λ0i

!

i  j=1

χ0j − E

H

&

i  j=1

√ + Δsρl W1l ; l = 1, · · · , n.

κj

'

eli



Δs (9.93)

Converting this equation to an expression in the state space, the next theorem shows that a full-factor simulation is achieved in the state space,

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eliminating the market price of risk. This result corresponds to Theorem 6.8.1 in the Gaussian HJM model. Theorem 9.7.1 For a full-factor LIBOR market model, let λl be the lth volatility component, defined by λl0i = ρl eli for i = 1, · · · , n. Let ϕ be the market price of risk obtained by applying Theorem 9.2.1. For an arbitrary initial forward LIBOR Li (0) and a small time step Δs > 0 with Δs < T1 , Li (Δs) is simulated by log Li (Δs) = log L1 (0) + −E H

&

i  j=1

 κj

E

H

&

' 

˜i ΔK Δt

'

Δs +

+ λ0i



!

i 

χ0j

j=1

Δsλ0i W1 .

(9.94)

Proof By the same argument as in Appendix B, we can convert equation (9.93) to the corresponding expression in the state space. Specifically, leftmultiplying both sides of equation (9.93) by (e1 , · · · , en ), the left-hand side becomes ⎛ n ⎛ ⎞ ⎞ 1 log L log L (Δs)e (Δs) i 1 i i=1 ⎜ ⎜ ⎟ ⎟ .. .. (e1 , · · · , en ) ⎝ (9.95) ⎠ =⎝ ⎠, . . n n log Ln (Δs) i=1 log Li (Δs)ei and the first term in the right-hand side becomes ⎛ ⎞ ⎛ n ⎞ 1 log L1 (0) i=1 log Li (0)ei ⎜ ⎜ ⎟ ⎟ .. .. (e1 , · · · , en ) ⎝ ⎠ =⎝ ⎠. . . n n log Ln (0) i=1 log Li (0)ei

(9.96)

From equation (9.16), the second term of equation (9.93) becomes ⎛ ⎞ - / 0. n i i H H 1 E [α ˜ 1 ] + i=1 λ0i ei j=1 χ0j − E j=1 κj ⎜ ⎟ ⎟ . 1 n ⎜ .. (e , · · · , e ) ⎜ ⎟ / 0. ⎝ ⎠    i i H n ˜ n ] + ni=1 λ0i χ − E κ e E H [α 0j j i j=1 j=1 ⎛ / - / 0. ⎞ 0 1 1 H ˜ 1 /Δt + λ01 E H ΔK j=1 χ0j − E j=1 κj ⎜ ⎟ ⎜ ⎟ .. =⎜ ⎟. . / - / 0. ⎠ 0 ⎝ n n H ˜ n /Δt + λ0n E H ΔK j=1 χ0j − E j=1 κj

(9.97)

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For the last term of (9.93), we have, from (9.73), that ⎛ ⎞ ⎛ n ⎞ 1 ρ1 W11 λ W e 0i 1 i i=1 ⎜ ⎟ ⎟ .. .. 1 n ⎜ (e1 , · · · , en ) ⎝ ⎠ = (e , · · · , e ) ⎝ ⎠ . . n n n ρn W1 i=1 λ0i W1 ei ⎛ ⎞ λ01 W1 ⎜ ⎟ .. = ⎝ ⎠. . λ0n W1

(9.98)

Summing each ith component separately, we obtain ' &  ˜i Δ K log Li (Δs) = log Li (0) + E H Δt ! i & i '    √ H +λ0i χ0j − E κj Δs + Δsλ0i W1 , j=1

j=1

(9.99)

for all i. This completes the proof. ✷ Properties of drift term Now, we can examine the properties of the drift term in the simulation model (9.94). In the Gaussian HJM model, we see, from equation (6.87), that the drift coefficient of the real-world model is simply given by E H [ΔFi /Δt]. Then, the drift is precisely the same as the rolled trend of the forward rate. In contrast, the drift term in the LMRW is somewhat different from equation (6.87). Specifically, the drift coefficient for the process log Li is expressed, from equation (9.94), as & ' ! i & i '   ˜i Δ K EH + λ0i χ0j − E H κj . (9.100) Δt j=1 j=1 Here, the first term indicates the rolled trend of Li in the sample period. Let us examine the properties of the second term. To address this, we note that we have already studied the case in which the initial LIBOR is given by the sample mean, as assumed in equation (9.85), Lj (0) = E H [K( ·, δi)].

(9.101)

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From the approximation (9.88), specifically, ' & i i   κj , χj (0) ≈ E H j=1

(9.102)

j=1

and the second term in equation (9.100) is approximately equal to zero. Thus, the drift is approximately equal to the rolled trend of K( ·, xi ). The following corollary summarizes this observation; this gives a state space expression of the form (9.91) in the PCA setup. Corollary 9.7.2 In a full-factor simulation model in the LMRW, if an initial forward LIBOR is equal to the sample mean of the historical forward LIBOR, as in equation (9.101), then it holds for i = 1, · · · , n that   √ H ΔKi log Li (Δs) ≈ log Li (0) + E (9.103) Δs + Δsλi Z(1). Δt

Since the term E H [ΔKi /Δt] represents the historical rolled trend of the forward LIBOR change, this corollary shows that real-world simulations are approximately equivalent to an empirical term structure model when the initial LIBOR is given by the sample mean. Drift with general initial conditions Next, let us consider a general case for the initial LIBOR and examine the properties of the drift term. For this objective, we develop our argument by using numerical approximation in a practical manner. If the initial LIBOR is higher than the sample mean, so that Lj (0) > H E [K( ·, δi)], then it roughly holds that ' & i i   κj . χj (0) > E H (9.104) j=1

j=1

Hence, the drift in equation (9.94) is roughly higher than the rolled trend. Conversely, if Lj (0) < E H [K( ·, δi)], then the drift will tend to be lower than the rolled trend. The difference from the case in the Gaussian HJM model is noteworthy. Building on this observation, we note that if the initial LIBOR curve is steeper than the sample mean of the LIBOR curve, then the simulation results in steepening drift, and similarly for higher-order observations.

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It might be fruitful to investigate the meaning of this feature from the viewpoint of real-world simulation or risk management in the future. Arbitrage pricing by simulation For arbitrage pricing by simulation, let us recall the framework for this from Section 5.3, where the implied short rate μ ¯(t) was defined by μ ¯(t) = μ ¯(Tm(t) ) such that μ ¯(Ti ) =

1 δi−1

log{1 + δi−1 Li−1 (Ti−1 )} ; i = 1, · · · , n

(9.105)

for each i. Here, μ ¯(t) is constant on each period (Ti−1 , Ti ] with i = 1, · · · , n. By using the market price of risk ϕ and the implied short rate μ ¯(t), the state price deflator was given, in equation (5.24), as   t    t |ϕ|2 ξ(t) = exp − ds − μ ¯(s) + ϕs dWs . (9.106a) 2 0 0 By applying the Euler approximation, we obtain the discretization of ξ(Δs) as     √ |ϕ|2 ξ(Δs) = exp − μ ¯(0) + (9.106b) Δs − ΔsϕW1 . 2 Thus, arbitrage pricing can be carried out by simulation in the LMRW. As an example, consider a security whose payoff at T < T1 is given by a ran1 dom variable X. The price pt of this security at t is given by pt = ξ(t) Et [ξ(T )X]. It is numerically possible to simulate the evolution of the LIBOR for longer periods, For this, we have two major approaches; one is using a timehomogeneous volatility, and the second is using a maturity-dependent volatility. It is a subject for future research to investigate what volatility structure is reasonable to use for long-period real-world simulation. In contrast, for a strictly numerical technique, see Norman (2009). 9.8

Dimensionality Reduction

We worked with the full-factor simulation model in the LMRW in Sections 9.6 and 9.7. For practical use, this section investigates how to reduce the number of factors needed for approximating the full-factor model.

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In the Gaussian HJM model, the dimensionality could be determined by measuring the accumulated contribution rate, as stated in Section 6.8. In this way, PCA mostly provides a solution to the problem of reducing dimensionality. However, as described in the following subsections, the simulation model in the LMRW has specific properties that PCA cannot capture. Thus, we need to introduce other measures for dimensionality reduction. Toward this objective, this section is divided into three subsections. Section 9.8.1 presents a setup that can be used to formulate the problem of dimensionality reduction. With this setup, Theorem 9.8.2 provides a fundamental result for reducing the number of dimensions. Furthermore, this theorem is modified to obtain Corollaries 9.8.3 and 9.8.4 in Section 9.8.3, which are practically useful. The argument here is just a numerical analysis problem. The proofs of Proposition 9.8.1 and Theorem 9.8.2 are presented in Appendix D. 9.8.1

Setup of Dimensionality Reduction

This subsection formulates the problem of dimensionality reduction in the LMRW. The basic setup is the same as in Sections 9.1 and 9.2. Additionally, we assume that the eigenvalues satisfy (ρ1 )2 > (ρ2 )2 > · · · > (ρn )2 > 0,

(9.107)

lim(ρl )2 = 0.

(9.108)

and l→n

Here, convergence is measured in the sense of engineering because {ρi }i is a finite sequence, and “convergence” will be used in the above sense in this section. In this context, we say that the accumulated contribution rate Cl satisfies liml→n Cl = 1. Let us recall the formulation of LIBOR process in the LMRW, given by equation (9.2) as ⎫ ⎧ ⎛ ⎞  t i ⎬ ⎨ t 2  |λi | ⎠ ⎝ Li (t) = Li (0) exp dt + λi dWt χj + λi ϕ − λi ⎭ ⎩ 0 2 0 j=m(t)

(9.109)

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for i = 1, · · · , n, where Wt denotes the n-dimensional Brownian motion under P. The d-factor model Our objective is to determine the number of factors d such that the d-factor model sufficiently approximates the full-factor model. As is conventional, let us begin our argument with the full-factor model. For a short time interval Δs > 0, the Euler approximation implies log Li (Δs) from equations (9.1) and (9.109) as n i n    δλl0j Lj (0) (λl0i )2 Δs − Δs log Li (Δs) = log Li (0) + λl0i 1 + δL (0) 2 j j=1 l=1 l=1 +

n 

λl0i ϕl Δs +

l=1

n 

√ λl0i ΔsW1l .

(9.110)

l=1

To avoid confusion, in the first d-factor model we denote Li , ϕl and γi by ˘ i , ϕ˘l and γ˘i , respectively. Hence, log L ˘ i (Δs) is represented as L ˘ i (Δs) = log Li (0) + log L

d 

λl0i

l=1

+

d 

λl0i ϕ˘l Δs +

l=1

d i   δλl0j Lj (0) (λl0i )2 Δs − Δs 1 + δLj (0) 2 j=1 l=1

d 

√ λl0i ΔsW1l .

(9.111)

l=1

With these, our objective is then simplified to the problem of how to determine the lowest dimensionality d such that equation (9.111) approximates equation (9.110) sufficiently well. 9.8.2

Dimensionality Reduction

This section investigates a method to reduce the dimensionality for simulation. For this, a measure, the “MPR score” (market-price-of-risk score) is introduced. First, we show the convergence of the market price of risk in the following proposition. Proposition 9.8.1 If the dimensionality d is high enough that Cd ≈ 1, then it follows that ϕ˘l ≈ ϕl for all l = 1, · · · , d.

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Proof See Appendix D.1. This proposition means that we must ensure the dimensionality d is sufficiently high even when we calculate only the first market price of risk. MPR score The n-dimensional constant vector γ = (γ1 , · · · , γn )T was defined, by equation (9.46), as & & i ' '  ˜i |λ0i |2 Δ K γi = E H κj + − λ0i E H . (9.112) Δt 2 j=1 By using this, the lth market price of risk was represented in equation (9.51)  as ϕl = ρ1l ni=1 γi eli . We define a constant vector ζ = (ζ1 , · · · , ζn )T by letting ζl =

n  i=1

γi eli ; l = 1, · · · , n.

(9.113)

With these, the market price of risk is represented by ϕl = ζl /ρl . Here, ζl is the lth projection of γ and also the score of its lth principal component. Along these lines, we call ζl the lth MPR score. MPR score in d-dimensional model The d-dimensional version of γi is denoted γ˘i . This is specified as ' & i ' &  ˜ |λ0i |2 H H Δ Ki − λ0i E κj + ; i = 1, · · · , n. γ˘i = E Δt 2 j=1

(9.114)

Here, the inner product in the second and third terms is defined in the ddimensional Euclidean space. Additionally, the MPR score ζ˘l for the d-dimensional model is defined by n  ˘ ζl = (9.115) γ˘i eli ; l = 1, · · · , d, i=1

where the inner product in the right-hand side is calculated in the same way as above. Then, the market price of risk for the d-dimensional model is represented by ζ˘l ; l = 1, · · · , d. (9.116) ϕ˘l = ρl

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The following theorem shows that the MPR score acts as a measure to determine the appropriate dimensionality for the simulation. Theorem 9.8.2 If the factor number d is sufficiently large such that Cd ≈ 1, and the absolute value of the MPR score |ζl | is sufficiently small for all l with d < l ≤ n, then the first d-factor model (9.111) approximates the full-factor model (9.110). Proof See Appendix D.2. ✷ Thus, the dimensionality for simulation is determined from a combination of the accumulated contribution rate and the MPR score. Remark We assumed that the volatility risk ρl is strictly decreasing in l, from equation (9.107). We assumed, also, that the market price of risk has the form ϕl = ζl /ρl ; here, ρl appears as a denominator. This means that |ϕl | may not be monotonically decreasing in l. Accordingly, the convergence of liml→n λl0i ϕl = 0 holds independently from the convergence of limd→n Cd = 1. This is a feature of dimensionality reduction that is intrinsic to the LMRW. There may be a different and simpler approach for reducing dimensionality, and study of this is suggested for future research. 9.8.3

Dimensionality Reduction in Practice

In practice, it is not feasible to compute the market price of risk ϕ and the MPR score ζ in the full-factor model. To accommodate practical use, we relax Theorem 9.8.2 as follows. Corollary 9.8.3 (Dimensionality reduction) For sufficiently large values of n ˜ ≤ n, let ζ˘l l = 1, · · · , n ˜ denote the MPR scores in the n ˜ -factor model. If ˘ Cd ≈ 1, and |ζl | is sufficiently small for all l with d < l ≤ n ˜ , then the first d-factor model (9.111) sufficiently approximates the full-factor model (9.110). In the following, the symbol “˘” is omitted for brevity. From the same observation as above, Theorem 9.7.1 approximately holds for the d-factor model, as the following corollary asserts.

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Corollary 9.8.4 Under the assumptions of Corollary 9.8.3, the following form in the d-factor model approximates the full-factor model in the formulation (9.94): ' & ˜i Δ K Δs log Li (Δs) = log Li (0) + E H Δt  i & i '   +λ0i χ0j − E H κj Δs √

j=1

j=1

+ Δsλ0i W1 ; i = 1, · · · , n,

(9.117)

where the third, fourth, and fifth terms are calculated in the d-factor model. When we intend to simulate the interest rate scenario only without pricing, it is sufficient to use the above form. For this, we need not use the market price of risk in the formulation. In particular, the following corollary provides an approximation of equation (9.91) in the state space. Corollary 9.8.5 If the initial LIBOR Li (0) is equal to the sample mean of the corresponding LIBOR, that is, if Li (0) = E H [K( ·, δi)], then the following approximation holds in the above d-factor model:   √ H ΔKi Δs + Δsλ0i Z(1) log Li (Δs) ≈ log Li (0) + E Δt

(9.118)

(9.119)

for i = 1, · · · , n.

This corollary states that even when we work with the d-dimensional model, the real-world simulation is approximately equivalent to a historical term structure so long as we have initial conditions satisfying equation (9.118) for each i. 9.9

Numerical Procedures in LMRW

This section describes a numerical procedure that can be used to build a simulation model in the LMRW. Let δ > 0 be fixed and let xi be a time

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length to the bond maturities such that xi = δi for i = 1, · · · , n. For a fixed time interval Δt > 0, we set the sequence of observation dates by letting tk = Δtk, k = 1, · · · , J + 1. ˜ i (tk ) Numerical calculation of ΔK ˜ i (tk ) was defined in equation (9.13) by ΔK ˜ i (tk ) = log K(tk+1 , xi − Δt) − log K(tk , xi ) ΔK

(9.120)

for i = 1, · · · , n, k = 1, · · · , J. As stated in Section 6.9 in the Gaussian HJM framework, we usually observe the forward LIBOR in the form K(tk , xi ) for a fixed time length xi at each tk . This means that we cannot directly observe K(tk+1 , xi − Δt), which appears in the first term of equation (9.120). To estimate K(tk+1 , xi − Δt), the conventional method uses K(tk+1 , xi ) as a proxy. Because the rolled trend explains the majority of the market price of risk, as mentioned in Section 9.5, it is recommended that we estimate the value by some type of interpolation. Similarly to the interpolation in equation (6.93), we here use linear interpolation, in the form   Δt Δt K(tk+1 , xi − Δt) = 1 − K(tk+1 , xi−1 ). (9.121) K(tk+1 , xi ) + δ δ

Naturally, spline interpolation or log-scale interpolation might provide a more reasonable estimate, but this is a matter for financial engineering. The significance of this linear interpolation will be shown by a numerical example in Section 10.2.3, where we will see that the specific properties of the real-world simulation are easily missed if we use K(tk+1 , xi ) instead of K(tk+1 , xi − Δt).

Calculation of the market price of risk Assume that the covariance matrix has rank n. We describe the procedure to estimate the market price of risk in the state space setup, as follows. 1) We calculate K(tk+1 , xi −Δt) by the interpolation (9.121). By using this, ˜ i (tk ) is obtained from equation (9.120) for i = 1, · · · , n, k = 1, · · · , J. ΔK 2) We calculate the eigenvalues (ρl )2 , the principal components el , and the volatilities σ l for l = 1, · · · , n by PCA on the covariance

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˜ i , ΔK ˜ j )/Δt. Cov(ΔK 3) The lth volatility component λl is determined by λli = ρl eli for i, l = 1, · · · , n. / 0 ˜ i is obtained from the sample data for i = 1, · · · , n. 4) E H ΔK

5) We fix a sufficiently large integer n ˜ with n ˜ ≤ n, and then calculate the values in further steps in the n ˜ -factor model. 6) We calculate the sequence κj (tk ), k = 1, · · · , J of equation (9.5) from the observation K(tk , δi) as κlj (tk ) =

λl0j δK(tk , δj) ; l = 1, · · · , n ˜ , j = 1, · · · , n. 1 + δK(tk , δj)

7) We calculate E H above.

(9.122)

/ i

0 l κ ˜ and i = 1, · · · , n from the j=1 j for l = 1, · · · , n

8) From equation (9.114) for the n ˜ -factor model, γi is obtained for i = 1, · · · , n by &

˜i ΔK Δt

'

&

i 

'

|λ0i |2 2 j=1 ' & & i '  n ˜ n ˜   ˜i (λl )2 ΔK H l H l − λ0i E κj + l=1 0i , = E Δt 2 j=1 l=1

γi = E

H

− λ0i E

H

κj +

(9.123)

where the inner product in the second and third terms is calculated in the n ˜ -factor model. 9) The MPR score ζl is obtained from equation (9.115) as ζl = for l = 1, · · · , n ˜.

n

i=1

γi eli

10) From equation (9.116), the market price of risk is obtained in the n ˜ -factor ˜. model as ϕl = ζl /ρl for l = 1, · · · , n The differences between this procedure and that in the Gaussian HJM framework are the following.

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• The dimensionality n ˜ is prefixed to estimate the market price of risk and the MPR score. • The procedures from steps 6) to 8) above are more complicated than the corresponding procedures for the Gaussian HJM model. Real-world simulation We next summarize the procedure to build a single-period simulation model by using Corollary 9.8.4, where the market price of risk does not appear explicitly. Additionally, we show a simulation model in the original form (9.68), which involves the market price of risk. 1) The procedures from steps 1) to 9) in the above are the same. 2) We calculate the accumulated contribution rate Cd by letting d (ρl )2 ; d = 1, · · · , n ˜. Cd = nl=1 2 l=1 (ρl )

(9.124)

3) By Corollary 9.8.3, the dimensionality d, d ≤ n ˜ is determined such that Cd ≈ 1 and the MPR score |ζl | is sufficiently small for all l with d < l ≤ n ˜. 4) We set the initial LIBOR Li (0) and generate the d-dimensional sequence of standard normal random numbers. 5) From equation (5.9), we compute χj (0) by letting χlj (0) =

λl0j δLj (0) ; l = 1, · · · , d, j = 1, · · · , n. 1 + δLj (0)

(9.125)

6) For a time period Δs with Δs < x1 , Li (δ) is simulated in a d-factor model by using Corollary 9.8.4. Specifically, ' &  i d   ˜i Δ K Δs + log Li (Δs) = log Li (0) + E H χl0j λl0i Δt j=1 l=1 ' & i d  √  l l Δs + Δs κlj −E H λ0i W1 (9.126) j=1

for i = 1, · · · , n.

l=1

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7) Alternatively, we may use, from equation (9.3), the following form instead of the above: ⎧ ⎫ i ⎨ 2⎬  |λ0i | log Li (Δs) = log Li (0) + λ0i χ0j + λ0i ϕ − Δs ⎩ 2 ⎭ j=m(t) √ + Δsλ0i W⎧ 1 d d i ⎨   l l λ0i χ0j + λl0i ϕl = log Li (0) + ⎩ l=1 l=1 j=m(t)  d d  |λl |2 √  l l 0i Δs + Δs (9.127) λ0i W1 , − 2 l=1 l=1 for i = 1, · · · , n. On the one hand, the formulation (9.126) is convenient because it highlights the features of the drift in the simulation. On the other hand, the formulation (9.127) is simple to use because it is immediately constructed from the spot LIBOR model. Arbitrage pricing Arbitrage pricing in the LMRW was addressed at the end of Section 9.7, where the implied short rate μ ¯(0) at t = 0 was given by μ ¯(0) = 1δ log{1 + δL0 (0)}. From this, the state price deflator is obtained from equation (9.106b) as     √ |ϕ|2 Δs − ΔsϕW1 . ξ(Δs) = exp − μ ¯(0) + (9.128a) 2 Specifically, & 

ξ(Δs) = exp − μ ¯(0) +

d

2 l=1 (ϕl ) 2



Δs −



Δs

d  l=1

'

ϕli W1l .

(9.128b)

By using this, arbitrage pricing can be executed. Long-term simulation By repeatedly carrying out single-period simulations, it is possible to simulate

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long-term evolution of the forward LIBOR and the state price deflator. Naturally, we may change the volatility structure and the market price of risk with the passage of time. This might be an interesting subject for future research.

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Chapter 10 NUMERICAL EXAMPLES

Abstract: This chapter presents numerical examples of real-world modeling. First, we give examples within the Gaussian HJM model, working with the Japanese LIBOR swap market data. We calculate the market price of risk from the data, referring to the interpretation of the market price of risk given in Chapter 6. After that, we show numerical examples in the LMRW in parallel to the above. Since the simulation model in the LMRW is more complicated than that of the Gaussian HJM model, we consider four different cases in the LMRW to clearly illustrate the properties of the real-world model. In these, Sections 10.1 and 10.2 present detailed examinations of the examples given in Yasuoka (2015) and Yasuoka (2012, 2013a), respectively. Next, we present an actual example that admits a positive market price of risk. For this, we employ the Hull–White model, working with data on U.S. Treasury yields. Also, working with long-period observations of U.S. Treasury yields, we calculate the market prices of risk in the Hull–White model. With this, we verify that long-period observation tends to cause a negative market price of risk. We examine the mean price property of the market price of risk by using U.S. Treasury yields. Additionally, Section 10.6 examines the properties of credit exposure calculation in connection with real-world modeling. These examples, in Sections 10.3, 10.4, 10.5, and 10.6 are original to this book.

Keywords: Convexity, Counterparty credit risk, Dimension reduction, Expected exposure, Flat yield model, Gaussian HJM model, Hull–White model, Interest rate swap, LIBOR market model, Market price of risk, Mean reversion, Mean price property, Monte Carlo simulation, Negative price tendency, Positive slope model, Potential future exposure, Real-world simulation, Swap rate, U.S. Treasury.

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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Valid for risk management Short rate model

HJM model

LIBOR market model

Gaussian HJM model CIR model Vasicek model CKLS model

Hull-White model Extended CIR model

Ho-Lee model

Figure 10.1: Inclusion relation among term structure models. A framework for real-world modeling has been developed for the models listed in the shaded regions.

Fig. 10.1 illustrates the inclusion relation among term structure models from the viewpoint of mathematical modeling1 . Some of the short rate models are not arbitrage-free in the sense of Section 4.4; that is, it is impossible to fit a term structure with arbitrary initial forward rates to those models. This figure shows that the HJM model and the LIBOR market model are appropriate for use in risk management. In the HJM model, what we can construct as a realworld model is restricted to the Gaussian HJM model. Along this line, we will use the same set of historical data to exemplify the real-world modeling of these types in Sections 10.1 and 10.2. The Ho–Lee model is a special case of the Hull–White model, in which the mean reversion rate vanishes. Specifically, the Ho–Lee model cannot be used when it is necessary to represent the mean reversion property. Because of this, the model has not been successfully used in recent practice. Here, we do not work with the Ho–Lee model when showing numerical examples. As the simplest model in this context, the Hull–White model is used in presenting an instance of a positive market price of risk and an instance of negative price tendency in Sections 10.3 and 10.4, respectively. 1

For details of the extended CIR model, see Hull and White (1990) or Munk (2011).

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Numerical Examples 0.03 0.025 0.02

0 2 5 10

0.015 0.01 0.005

Period A

Period B

0

Figure 10.2: Forward rates in the Japanese LIBOR swap market. The implied six-month forward rate is regarded as the instantaneous forward rate. Period A is from 2 April 2007 to 16 June 2008, and Period B is form 16 June 2008 to 31 August 2009. Swap rate data were provided by Mizuho Information and Research Institute. LIBOR data were obtained from Riskmatrix (2014). Implied forward rates were calculated by the author.

10.1

Real-world Model in the Gaussian HJM Model

This section presents numerical examples of real-world modeling within the Gaussian HJM framework. We verify the numerical properties of the market price of risk and the drift in the simulation, referring to the results in Chapter 6. 10.1.1

Estimation of Market Price of Risk

Sample data description We use the Japanese LIBOR swap rates for April 2007 through August 2009 and apply the cubic spline algorithm to interpolate the interest rates at semiannual intervals. We set δ = 0.5 (year) and solve for the six-month forward rate by bootstrapping. For convenience, we regard the six-month forward rate as the instantaneous forward rate.

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0.03 0.025 0.02

Period A

0.015 Period B

0.01

4/2/2007 6/16/2008

0.005

8/31/2009

0 0

1

2

3

4

5

6

7

8

9

10

Term to maturity (year)

Figure 10.3: Forward rate curves at 2 April 2007, 16 June 2008, and 31 August 2009. Period A is from 2 April 2007 to 16 June 2008, and Period B is from 16 June 2008 to 31 August 2009.

Fig. 10.2 exhibits a historical chart of the forward rates with maturities of 0, 2, 5, and 10 years. This chart shows a sudden fall of the forward rates in early October 2008; that fall was related to the huge decline in the stock market as a consequence of the 2008 global financial crisis. Proposition 6.5.1 states that the market price of risk is basically determined by the rolled trend, and so from this observation it is expected that this sudden change would have little influence on the market price of risk. To examine the numerical properties of the market price of risk, we split the sample period into two periods. Period A is earlier and covers 2 April 2007 to 16 June 2008 (297 data points), where the forward rates show an upward trend. Period B is later and covers 16 June 2008 to 31 August 2009 (298 data points), where the forward rates show a downward trend. Fig. 10.3 presents the forward rate curves of three specific days (2 April 2007, 16 June 2008, and 31 August 2009). These curves show that the observable trend is bear-flattening in period A and bull-steepening in period B. Fig. 10.4 presents the sample mean of the forward rates in each period, where “mean ± σ” represents the mean plus/minus one standard deviation of the sample rates. We can see that the forward rate curve is mostly upwardsloping in Period A.

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Numerical Examples 0.03 0.025 0.02 0.015 0.01

Mean + σ Mean Mean - σ

0.005 0 0

1

2

3

4

5

6

7

8

9

10

Term to maturity (year) Period A 0.03 0.025 0.02 0.015 0.01

Mean + σ Mean Mean - σ

0.005 0 0

1

2

3

4

5

6

7

8

9

10

Term to maturity (year) Period B

Figure 10.4: Mean and standard deviation of sample forward rates In Period B, however, the forward rate curve is downward-sloping at the short end2 , and upward-sloping at maturity of 1 to 10 years. Because of this, the forward rates roughly admit a roll-up tendency at the short end and a roll-down tendency for maturity of 2 to 10 years.

2

Precisely speaking, the spot LIBOR refers to the LIBOR itself, and the forward rate is implied from the LIBOR and swap rates. Because of this relation, this negative slope at the short end might be due to a difference of markets rather than due to the term structure of the interest rates.

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0.8 0.6 0.4 0.2 i

0 1

3

5

7

9

11

13

15

17

19

-0.2 -0.4

Period A 1 2 3 4

0.6

0.4

0.2 i

0 1

3

5

7

9

11

13

15

17

19

-0.2

-0.4

Period B

Figure 10.5: First four principal components for Period A and Period B Calculation of the market price of risk We set Δt = 0.08 (equivalent to 20 days). According to the procedure in Section 6.9, we compute the market price of risk. The value σ0i υ0i in equation (6.97) is approximately calculated in the eight-factor model. Table 10.1 shows the eigenvalues (ρl )2 , the contribution rates, and the market prices of risk ϕl for each component. We can verify that the size of ϕl is not decreasing in l, as remarked in the proof of Theorem 9.8.23 . Specifically, |ϕl | attains a maximum at l = 6 in Period A and at l = 5 in Period B. The accumulated contribution rate shows that the first three components 3

Although this remark was made in regard to the LMRW, it is obviously valid for the Gaussian HJM model as well.

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0.004 0.002 0 -0.002 -0.004 E(ΔF㸧 συ

-0.006 -0.008

i

-0.01 1

3

5

7

9

11

13

15

17

19

Period A

0.003 0.001 -0.001 -0.003 -0.005 -0.007

E(ΔF㸧 συ

-0.009

i

-0.011 1

3

5

7

9

11

13

15

17

19

Period B

Figure 10.6: Two component terms of γ for Period A and B. γ is represented by γi = E H [∆Fi /∆t] + σ0i υ0i . In the graph, E(∆F ) and συ denote E H [∆Fi /∆t] and σ0i υ0i , respectively, and E(∆F ) shows the term structure of the rolled trend.

explain more than 98% of the covariance. Fig. 10.5 exhibits the first four principal components for both periods. As mentioned in Section 4.3, the first three components represent, respectively, parallel shifts, steepness changes, and curvature changes of the forward rate curve.

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Table 10.1: Eigenvalues and the market price of risk. Real-world modeling in the Gaussian HJM model.

l 1 2 3 4 5 6 7 8

Period A (3 April 2007 to 16 June 2008) Eigenvalue Contribution Accumulated Market price rate contribution rate of risk 2 Cl ϕl (ρl ) 7.61E-04 0.9305 0.9305 -0.256 4.53E-05 0.0554 0.9859 0.532 5.44E-06 0.0067 0.9926 0.598 3.29E-06 0.0040 0.9966 0.495 1.22E-06 0.0015 0.9981 0.896 5.86E-07 0.0007 0.9988 -1.049 3.22E-07 0.0004 0.9992 -0.011 2.45E-07 0.0003 0.999 0.311

l 1 2 3 4 5 6 7 8

Period B (16 June 2008 to 31 August 2009) Eigenvalue Contribution Accumulated Market price rate contribution rate of risk 2 Cl ϕl (ρl ) 4.77E-04 0.8082 0.8082 -0.884 8.35E-05 0.1413 0.9495 -1.886 2.38E-05 0.0402 0.9897 -0.816 3.14E-06 0.0053 0.995 1.113 1.31E-06 0.0022 0.9972 2.445 5.57E-07 0.0009 0.9982 -0.557 3.36E-07 0.0006 0.9988 1.746 2.55E-07 0.0004 0.9992 -0.709

Market price of risk and the historical trend Let us verify the interpretation of the market price of risk that was given in Section 6.5. In equation (6.29), γi was defined as γi = E H [ΔFi /Δt] + σ0i υ0i .

(10.1)

Additionally, Proposition 6.5.1 assumes that the greater part of γi is from E H [ΔFi /Δt], and specifically |σ0i υ0i | ≪ E H [ΔFi /Δt]. In this respect, Fig. 10.6 shows the behavior of E H [ΔFi /Δt] and σi υi (abbreviated as E(ΔF ) and συ, respectively, in the figure). We see that |E H [ΔFi /Δt]| is considerably

Numerical Examples

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larger than |σi υi | in both periods. From this, it is reasonable to accept that the above inequality is almost satisfied in these cases. Along these lines, the market price of risk is approximately explained by the rolled trend from Proposition 6.5.1, as well as roughly explained by the observable trend in both cases. The observable trend is almost bear-flattening in period A, and the row labeled “bear flat” in Table 6.4 shows that ϕ1 ought to be near zero and ϕ2 should be positive. Table 10.1 shows that ϕ1 = −0.256,

ϕ2 = 0.532

(10.2)

in this period, and so they are roughly explained by Table 6.4 . The observable trend of period B is bull-steepening. For this, Table 6.4, in the row labeled “bull steep,” shows that ϕ1 and ϕ2 should both be negative. Table 10.1 shows that, in Period B, ϕ1 = −0.884,

ϕ2 = −1.886.

(10.3)

These values are also consistent with Table 6.4. From these, the market price of risk is mostly determined by the historical trend, rather than as an impact from the sudden fall of the interest rates in 2008. If possible to perform, the t-test for this estimate might provide useful information about these estimates (this is left as a future subject). We node that both Period A and Period B have a time length of around 300 business day, which is relatively short in the sense of a negative price tendency. Because of this, it is not asserted that the condition (6.60) is always satisfied. 10.1.2

Observation on Simulation

Dimension reduction Next let us determine the dimensionality such that the simulation model approximates the full-factor model. In the manner described in Section 6.8, it is sufficient to determine the number of factors d such that the accumulated contribution rate Cd is close to 1. Table 10.1 shows that C3 is greater than 0.98 for both periods. Hence, we work with a three-factor model for both Period A and Period B.

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Period B 0.03 0.025 0.02 0.015 0.01

Inial rates Risk-neutral Real-world

0.005 0 0

1

2

3

4

5

6

7

8

9

Term to maturity (year)

Period A 0.03 0.025 0.02 0.015 Inial rates Risk-neutral Real-world

0.01 0.005 0 0

1

2

3 4 5 6 7 Term to maturity (year)

8

9

Figure 10.7: Mean of real-world simulation and risk-neutral simulation at t = 0.5 (year) Drift in the real-world simulation Here, we consider a single-period simulation, using equation (6.91) in the three-factor model. The forward rate observed of 31 August 2009 is taken as the initial forward rate, which was shown in Fig. 10.3. The rolled trend E H [ΔFi /Δt] is exhibited in Fig. 10.6, where Period B shows a remarkable roll-down of short-term forward rates. In both cases, the negative slope of the forward rate curve is observed at the short end. To examine this feature further, let us compare real-world simulation with riskneutral simulation for the two cases. In this, the simulation associated with Period A (resp., Period A) will be called Case A (resp., Case B). Fig. 10.7 presents the means of the forward rate at t = δ for Case A and

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Term to maturity (year) Period B

Figure 10.8:

Simulated forward rates at t = 0.5 years; 50 simulations with a three-factor model

Case B. As explained by Fig. 7.1 in Section 7.1, the risk-neutral simulation shows that the mean of f (δ, Ti ) is almost equal to the parallel shift to the left in the initial rate f (0, Ti ) for Δs in both cases. This feature is typical of arbitrage-free term structure models. In contrast, the mean of the real-world simulation shows a different property, and so reproduces the rolled trend, as shown in Fig. 10.6. Along these lines, Fig. 10.7 provides a visualization of the difference between the real-world simulation and the risk-neutral one. A more detailed examination within the LMRW will be presented in Section 10.2.

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Monte Carlo simulation Following the numerical procedure described in Section 6.9, a Monte Carlo simulation is executed. Fig. 10.8 shows 50 simulations of f (δ, xi ) for each case, partially showing the negative values of short-term forward rates. Although the possibility of a negative interest rate is a drawback of the Gaussian model, the Gaussian HJM framework provides a perspective on the real-world model. However, negative yields have been recently4 observed in the government bond market of some European countries and of Japan. The HJM model may be valid for such markets. 10.2

LIBOR Market Model

This section presents numerical examples of simulation in the LMRW that parallel those in Section 10.1. In the modeling, the number of factors is determined to approximate the full-factor model by using Corollary 9.8.3, and the simulation model is constructed by using Corollary 9.8.4. We examine the simulation properties with a particular focus on the drift feature near the short end of the forward rate curve. 10.2.1

Estimation of Market Price of Risk

Sample data description We again use the Japanese LIBOR swap rates, as we did in Section 10.1. Accordingly, the earlier Period A covers 2 April 2007 to 16 June 2008, where the forward LIBOR shows a tendency to rise. The later Period B covers 16 June 2008 to 31 August 2009, where the forward LIBOR shows a tendency to fall, as seen in Fig. 10.2. We recall that the observable trend is bear-flattening in Period A and bull-steepening in Period B, as shown in Fig. 10.3. Market price of risk We set Δt = 0.08 (equivalent to 20 days). The procedure described in Section ˜ i , ΔK ˜ j ) from the sam9.9 is used to construct the covariance matrix Cov(ΔK 2 ple data. Table 10.2 shows the eigenvalues |ρl | , the accumulated contribution 4

This book was written over the period from summer 2014 to spring 2015.

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Table 10.2: Eigenvalues and the market price of risk in the LIBOR market model. ρ2 , Cl , ζl , and ϕl indicate the eigenvalue, the accumulated contribution rate, the MPR score, and the market price of risk, respectively. For comparison, ϕHJM indicates the market price of risk as obtained from the Gaussian HJM model for the same period.

l 1 2 3 4 5 6 7 8

Period A ρ2 2.615 0.2366 0.0284 0.0146 0.0059 0.0024 0.0011 0.0005

(3 April Cl 0.8999 0.9813 0.9911 0.9961 0.9982 0.999 0.9994 0.9996

2007 to 16 June ζl ϕl 0.061 0.038 0.369 0.758 0.126 0.746 0.014 0.114 0.016 0.205 0.061 1.226 -0.034 -1.017 0.004 0.155

2008) ϕHJM -0.256 0.532 0.598 0.495 0.896 -1.049 -0.011 0.311

Period B (16 June 2008 to 31 August 2009) l ρ2 Cl ζl ϕl ϕHJM 1 1.756 0.7191 -1.449 -1.094 -0.884 2 0.4904 0.9199 -1.203 -1.718 -1.886 3 0.1481 0.9805 0.082 0.213 -0.816 4 0.024 0.9904 0.398 2.57 1.113 5 0.01 0.9945 0.209 2.089 2.445 6 0.0091 0.9982 0.053 0.55 -0.557 7 0.0014 0.9988 -0.004 -0.115 1.746 8 0.0011 0.9992 0.071 2.113 -0.709 rates, the MPR scores ζl , and the market prices of risk ϕl for each compoAdditionally, nent, where ζl and ϕl are calculated in the 8-factor model. the rightmost column contains the market prices of risk that were estimated in the Gaussian HJM model for the same sample period and appeared in the rightmost column of Table 10.1. Table 10.2 shows that ϕ1 = 0.038 and ϕ2 = 0.758 in Period A. Since the observable trend is bear-flattening in this period, these values are consistent with Table 9.4 (in the row labeled “bear flat”), where ϕ1 is predicted to be near zero and ϕ2 is predicted to be positive. Period B in Table 10.2 shows that ϕ1 = −1.094 and ϕ2 = −1.718. Since

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the observable trend in this period is bull-steepening, these results are also explained by Table 9.4 (in the row labeled “bull steep”), where ϕ1 and ϕ2 are predicted to be negative. 10.2.2

Four Cases: Cases B1 to B4

This section begins with determining the dimensionalities for Period A and Period B. From this, we investigate the properties of the simulation for Period B, setting four cases for this period. Dimension Reduction Fig. 10.9 shows the first four principal components el , l = 1, · · · , 4 for periods A and B. Similarly to Section 10.1.1, the first three components explain the changes in the level, slope, and curvature, respectively, of the forward LIBOR curve. In both periods, the fourth component seems to affect the movement of the short-term forward rates. This observation is consistent with that in Longstaff et al. (2002). We determine the number of factors to use to approximate the full-factor model by applying Corollary 9.8.3. In Table 10.2, the accumulated contribution rate shows that the first three components explain more than 98% of the covariance for the two periods. Furthermore, both periods show a nondecreasing market price of risk in l, as mentioned in Section 9.8. Indeed, |ϕl | attains a maximum at l = 6 in Period A, and at l = 5 in Period B. In Period A, the MPR score ζl is relatively small for l > 3. Hence, a three-factor model is sufficient for a simulation. In Period B, the MPR score ζl is quite small for l > 6. Hence, a four- or five-factor model is necessary for simulations in Period B. From this, Period B is a distinct example of a case where the number of factors is determined by the MPR score, rather than by the PCA. Approximation of market price of risk We recall the definition of γi in equation (9.46) as & ' & i '  ˜i Δ K |λ0i |2 γi = E H − λ0i E H . κj + Δt 2 j=1

(10.4)

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Figure 10.9: The first four principal components Fig. 10.10 presents γi and three terms from the right-hand side of the above.  We see that the second term, λi E H [ ij=1 κj ], is negligibly small in both cases. ˜ i /Δt] In period A, |λi |2 /2 does not seem to be smaller than the term E H [ΔK for some i. Hence, the inequality (9.62) in Proposition 9.5.1 does not hold for this case. In contrast, it roughly holds for period B in Fig. 10.10, with & ' ˜i |λi |2 Δ K ≪ EH , (10.5) 2 Δt for almost all i, so the inequality (9.62) is almost satisfied. Then, the approximation of the market price of risk in Proposition 9.5.1 holds for Period B, as

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Period A

Period B

Figure 10.10: γ and its components for Period  A and Period B. E[∆K], λE[

and λ/2 are abbreviations of

˜ E H [∆K

i /∆t],

λi

EH [

i j=1 κj ],

and |λi

|2 /2,



κ], respectively.

˜ l /ρl . ϕl ≈ R

Case B1 to B4 Since Period B requires higher dimensional modeling than Period A does, it is expected that Period B will exhibit a characteristic particular to simulation in the LMRW. Because of this, we examine the simulation for Period B, working with the following four cases. • Case B1: Four-factor real-world simulation according to the numerical procedure in Section 9.9.

Numerical Examples

Table 10.3:

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Summary of cases for Period B

Case Dimensionality B1 4 B2 4 B3 3 B4 4

Measure K(tk+1 , δi − Δt) real-world interpolation risk-neutral interpolation real-world interpolation real-world K(tk+1 , δi)

• Case B2: Four-factor simulation under the spot LIBOR measure, executed by setting ϕ = 0 for Case B1. • Case B3: Three-factor real-world simulation based on Case B1. • Case B4: Four-factor real-world simulation, where K(tk+1 , δi) is used as ˜ i (tk ). a proxy of K(tk+1 , δi − Δt) to compute ΔK Case B1 mostly approximates the full-factor model, and is carried out in the manner described for modeling in Section 9.9. Then, we may regard Case B1 as a standard for comparison with other cases. Case B2 compares the risk-neutral simulation of the standard case of Case B1. Case B3 highlights things missed by the three-factor model. Case B4 examines the effect of the interpolation (9.121). Table 10.3 summarizes these cases to clarify the differences among them. 10.2.3

Examination of Four Cases

The implied forward LIBOR observed on 31 August 2009 is taken as the initial LIBOR, as in Section 10.1. In the four cases, Li (δ) is calculated by using a single-period simulation. We examine these cases by comparing the mean of each Li (δ). For convenience, we evaluated the mean in log-scale by using E[log Li (δ)] as a proxy of E[Li (δ)]. Here, E[log Li (δ)] is obtained, from equation (9.3), as   i  |λi |2 δ, (10.6) κj (0) + λi ϕ − E[log Li (δ)] = log Li (0) + λi 2 j=1 for the given L(0), λ, and ϕ.

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If we set ϕ = 0 and P∗ is the spot LIBOR measure, then P = P∗ holds from the argument in Section 9.9. As well, E[·] = E ∗ [·] holds for Case B2, where E ∗ [·] denotes expectation under P∗ . It holds that E ∗ [log Li (δ)] = E[log Li (δ)]|ϕ=0  = log Li (0) +

i 

|λi |2 λi κj (0) − 2 j=1



δ.

(10.7)

Then, E[log Li (δ)] is obtained for all four cases by the direct calculation of equations (10.6) and (10.7). Fig. 10.11 shows the mean E[log Li (δ)] against the initial forward LIBOR in log scale for each case. This graph gives interesting properties in the realworld simulation, which are described in the following. Case B1 This case mostly closely approximates the full-factor model among the four cases. The mean of the simulated forward LIBOR curve at t = δ shows a bull-steepening trend in Fig. 10.11 (Case B1). It is noteworthy that the mean of the curve at t = δ is downward-sloping at the short end. Case B1 is the only model that simulates this feature. Case B2 Case B2 is simulation under the spot LIBOR measure with the same parameters as in Case B1. From Fig. 10.11 (Case B2), the shape of the mean curve E[log Li (δ)] is almost equal to the initial curve after a parallel shift to the left for δ. This feature is typical of the risk-neutral model, as explained in Sections 7.1 and 10.1.2. Namely, it roughly holds that E[Li (δ)] ≈ Li (0) for all i. This is quite different from the result in Case B1. Naturally, E[log Li (δ)] has positive slope at the short end of the curve. We furthermore note that Case B2 has a bear-flattening tendency, which is also different from in Case B1. Hence, the difference between the real-world simulation and the risk-neutral simulation is observed not only in the level of the interest rates but also in the shape of the forward LIBOR curve.

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Case B1

Case B2

Case B3

Case B4

Figure 10.11: Initial forward LIBOR and the mean of simulation. The initial forward LIBOR is shown in log scale, and the dashed line shows the mean of the simulation, E[log Li (δ)].

Case B3 Case B3 is a three-factor simulation in the LMRW. The results shown in Fig. 10.11 (Case B3) are mostly similar to those of Case B1, except at the short end of the forward LIBOR curve. Indeed, E[log Li (δ)] in Case B3 is almost flat at the short end. This means that the three-factor model fails to simulate the short-end dynamics of the forward rate curve. From this, we may take the view that the fourth factor might affect the movement of the short-term forward LIBOR. Case B4 Case B4 uses K(tk+1 , δi) instead of K(tk+1 , δi − Δt) without interpolation by

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equation (9.121). Fig. 10.11 (Case B4) shows that the result seems to be intermediate between Case B1 and Case B2. Specifically, the entire forward LIBOR curve admits a bull-steepening trend like that seen in Case B1, and E[log Li (δ)] is upward sloping at the short-end. Hence, Case B4 misses both the effect of the fourth factor and the feature of the real-world model. Therefore, we may say that real-world simulation is necessary to properly estimate K(Δt, Ti − Δt). Summary of four cases Consequently, both Case B1 and Case B3 admit plausible results for real-world simulations. However, only Case B1 implies that the negative slope of the forward LIBOR curve at the short end will continue for a half-year (cf. the first footnote in Section 10.1.1). In this respect, Case B1 is suitable for risk management and for profit–loss simulation, where we forecast that the historical behavior of the LIBOR will continue to the future. Accordingly, Case B3 is the second best for this purpose. Finally, Fig. 10.12 shows 50 simulations of the forward LIBOR at t = δ in Case B1. Comparing this with Fig. 10.8, the LMRW simulates higher forward rates than the Gaussian HJM model does for the longer term. For valuing the interest-rate-rise risk, this model difference might not be negligible, and this should be studied further in regard to practice. 0.04

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Figure 10.12: Forward LIBOR at t = 0.5 years in Case B1, 50 simulations

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Remark on dimension reduction In the Gaussian HJM model, a three-factor model is sufficient to simulate the negative slope of the forward rate curve at the short end, as shown in Fig. 10.7 (Period B). In contrast, to simulate a similar feature in the LMRW, it is necessary to use at least four factors in the model, as shown by Case B1 and Case B3 together. This difference might be due to the complexity of the LMRW, rather than model selection. Indeed, the drift term in the Gaussian HJM model has a simple form, as E H [ΔFi /Δt]Δs, shown in equation (6.87), which is independent from the dimensionality. In the LMRW, the drift term is given by equation (9.127) as ⎫ ⎧ i ⎨ 2  |λ0i | ⎬ λ0i Δs, (10.8) χ0j + λ0i ϕ − ⎩ 2 ⎭ j=m(t)

and this term depends on the dimensionality. Let us recall the argument for dimension reduction from Section 9.8.2 and Appendix D.2, where the second term in the above was reduced to eli ζl for each lth component from equation (D.15). Specifically, we have λli ϕl = eli ζl . Here, ζl is the lth MPR score, which is the key to the dimension reduction. Thus, dimension reduction is not trivial in the LMRW, in contrast to the case for the Gaussian HJM model. 10.3

Positive Market Price of Risk

This section presents an actual example, in which the first market price of risk ϕ1 takes a positive value. For this objective, we employ the Hull–White model, showing the numerical procedure to construct a real-world model. As mentioned at the end of Section 7.1, the value of the first market price of risk ϕ1 depends on the rolled trend score. From this observation, the market price of risk does not necessarily take a negative value. Indeed, when we use only one-year observation data, it might not be difficult to find a period that admits a positive ϕ1 . However it turns out to be difficult to do that when the observation period is taken to be longer, which reflects the negative price tendency of the market price of risk.

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Figure 10.13: Implied forward rates and observable trends in the U.S. Treasury market. 0, 3, and 10 in the figure indicate the forward rates over the intervals [0, 0.5], [3, 3.5], and [10, 10.5] years, respectively. The yield data were retrieved from the Board of Governors of the Federal Reserve System (2014); in those data, monthly yield represents the monthly average of daily yields. The forward rates were calculated by the author. From the argument for the negative price tendency in Section 7.2, we may view this tendency as not being strongly affected by the choice of model. In this respect, the Hull–White model has been successfully used for studies on

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the term structure of interest rates, and that model is a special case of the Gaussian HJM model. Along these lines, we employ a one-factor Hull–White model for the examination. We use monthly U.S. Treasury yields from December 1958 to October 2012 and find the six-month implied forward rates. Fig. 10.13(a) shows a historical chart of the implied forward rates. The forward rates rise remarkably from the late 1970s to the early 1980s, reaching nearly 16% in the early 1980s. Hence, we expect that this period might admit a positive market price of risk. In the following calibration, we regard the six-month forward rate as the instantaneous forward rate for convenience. Detection of period with positive ϕ1 To find a period that admits a positive market price of risk, let us recall Proposition 6.5.1, and specifically that ϕ1 is approximately represented by ϕ1 ≈ R1 /ρ1 under certain conditions. Here, R1 is the first rolled trend score and (ρ1 )2 is the first eigenvalue. Using this approximation, we want to find a five-year period such that R1 > 0. However, it is not feasible to calculate the rolled trend score for each of the multitudinous five-year periods in the sample. For convenience, we examine the observable trend as a proxy for the rolled trend score. From equation (6.53), the observable trend for a five-year period [t, t + 5] is obtained as F (t + 5, x) − F (t, x) (10.9) 5 for each time t. We calculate this with x fixed at 0, 3, and 10 years for all t. Fig. 10.13(b) shows the observable trends for the sample period. We find that these trends take a nearly maximal value near March 1975. From this observation, it is expected that the first rolled trend score will take a positive value for the period from March 1975 to March 1980. Thus, we adopt this period for the examination. Volatility structure To estimate the market price of risk for the period above, let us begin by determining the volatility structure in the Hull–White model. We set δ = 0.5 years and xi = δi for i = 1, · · · , 20. Calculating ΔF (tk , xi ) by the interpola-

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Table 10.4: Rolled trend for the sample period (March 1975 to March 1980), and the Hull–White volatility. The interest rate dataset is the same as for Fig. 10.13.

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Rolled trend E H [ΔF/Δt] -0.0067 0.0164 0.0236 0.0096 0.0054 0.0066 0.0063 0.0057 0.0054 0.0052 0.0048 0.0046 0.0047 0.0053 0.0061 0.0068 0.0075 0.0079 0.0081 0.0080

Principal Hull–White component volatility ei σ exp(κxi ) 0.394 0.0176 0.423 0.0164 0.316 0.0152 0.250 0.0142 0.249 0.0132 0.248 0.0122 0.234 0.0114 0.216 0.0106 0.201 0.0098 0.188 0.0091 0.175 0.0085 0.163 0.0079 0.154 0.0073 0.147 0.0068 0.140 0.0063 0.135 0.0059 0.130 0.0055 0.126 0.0051 0.122 0.0047 0.119 0.0044

tion (6.93) for all k and i, we find the first principal component e1 and the eigenvalue ρ1 by PCA. As a result, we have (ρ1 )2 = 0.0022. Since we are working with a one-factor model, we may omit the “1” superscripts and subscripts (e1 , ρ1 , etc.) in this section. Table 10.4 shows the rolled trend E H [ΔFi /Δt], the principal component (e1 , · · · , e20 )T , and the Hull–White volatility σ exp(κxi ) for the sample data. Here ei , i = 1, · · · , 20 shows obvious decaying of the volatility structure, and specifically shows strong mean reversion. Applying the method of Section 8.4, we compute the parameters σ and κ. The results are as follows: σ = 0.0190,

κ = 0.146.

(10.10)

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Figure 10.14: The principal component volatility and the Hull–White volatility. The interest rate dataset is the same as for Fig. 10.13.

As expected from the above, the mean reversion coefficient κ is positive. Fig. 10.14 compares the observed volatility component (ρe1 , · · · , ρe20 )T with the Hull–White volatility σ exp(κxi ), i = 1, · · · , 20. We visually see that the Hull–White volatility well approximates the observation volatility. We note that the volatilities of short-term rates are almost all higher than 1.5%, decaying to near 0.5% for the long-term forward rates. Market price of risk The second column of Table 10.4 shows the rolled trends, which are positive for almost all i. It is expected that the first rolled trend score R will be positive. And, in fact the calculation of equation (8.30) implies that R = 0.0304. From equation (8.25), we obtain that β = −0.0020. Theorem 8.5.1 tells that the sign of the market price of risk depends on whether R > β. From this, the above calculation implies a positive market price of risk. Specifically, this is estimated to be positive as 0.0304 − 0.0020 √ = 0.609 (10.11) 0.0022 from Theorem 8.5.1. As intended, we have found a period that admits a positive market price of risk. ϕ=

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Figure 10.15: Expectation of {f (∆s, δi) − f (0, δi)} with ∆s = 1 (year) under

P and Q. “Real-world” indicates the expectation under P, and “Risk-neutral” indicates that under Q. The interest rate dataset is the same as for Fig. 10.13.

Real-world simulation Finally, we examine the difference between real-world simulation and riskneutral simulation. For a one-year simulation, by setting Δs = 1, we calculate the expectations of f (Δs, δi) in P and Q according to Corollary 8.5.2(1). These are compared in Fig. 10.15. The graph shows that the real-world model simulates higher forward rates than the risk-neutral model does. Of particular note, the short-term to middle-term forward rates are considerably higher than the corresponding rates in the risk-neutral model. This difference is evaluated by equation (8.38) in Corollary 8.5.2 (2) as E[f (Δs, δi)] − E Q [f (Δs, δi)] = σe−κδi ϕΔs ; i = 1, · · · , 20.

(10.12)

Here, δ = 0.5 (year), Δs = 1 (year), and the figures in equations (10.10) and (10.11) are substituted into σ, κ, and ϕ in the right-hand side of equation (10.12). This difference is plotted in Fig. 10.16, in which we see that the real-world model simulates a rate 100 to 25 basis points higher for the forward rates than the risk-neutral model does in expectation. Further, the difference is exponentially decaying in i, which is due to the structure of the right-hand side of equation (10.12).

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RW-RN 0.012 0.01 0.008 0.006 0.004 0.002 0 1

3

5

7

9

11

13

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19 i

Figure 10.16: Difference between the real-world simulation and the risk-neutral simulation. The graph plots the difference E[f (∆s, δi)] − E Q [f (∆s, δi)] at ∆s = 1 (year). The interest rate dataset is the same as for Fig. 10.13.

10.4

Negative Price Tendency

In the previous section, we estimated the market price of risk in the U.S. Treasury market, from which we found a period so that the first market price of risk takes a positive value. Using the same data, we estimate the market price of risk for a long-period observation and verify the negative price tendency. We refer back to Fig. 10.13(a), the chart of the implied forward rates found from the U.S. Treasury yields. From this dataset, we define three periods of twenty years, as follows. • Period A: March 1960 to March 1980. • Period B: March 1970 to March 1990. • Period C: March 1980 to March 2000. Period A is a term before the peak in the interest rate at the early 1980s; during this term, the interest rates are rising on average. Period B includes the peak at the middle of the period; during this term, the interest rates are

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March 1960 March 1970 March 1980 March 1990 March 2000

0.18 0.16 0.14 0.12

A

0.1

C

B

0.08 0.06 0.04 0.02 0 0

1

2

3

4

5

6

7

8

9

10

Figure 10.17: Forward rate curves for March of 1960, 1970, 1980, 1990, and 2000. The interest rate dataset is the same as for Fig. 10.13.

mostly rising until 1980 and falling after that. Period C is a term after the peak, during which the interest rates show a tendency to fall. Fig. 10.17 shows the forward rate curves for March of 1960, 1970, 1980, 1990, and 2000. The changes of the curves during each period are indicated by the labeled arrows. Estimation by flat yield model In Fig. 10.17, the graphs show that the forward rates are approximately flat. From this observation, we simply estimate the sign of the market prices of risk, working with a flat yield model to do so. Following the setup in Section 7.2, we set n = 20, δ = 0.5, and τ = 20 for all periods. We refer back to Fig. 10.14, which shows the volatility structure in connection with the Hull–White volatility for the period from 1975 to 1980. From this view, we assume that the volatility σ is flat, at 1.5%, through the entire period from 1960 to 2000. Referring to Theorem 7.2.1, the right-hand side of equation (7.15a) is calculated as δ(n − 1) 0.5 × 19 (σ(τ ))2 = (0.015)2 = 0.00107. 2 2

(10.13)

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Here, we let σ(τ )=0.015. From the assumption of constant volatility over the whole period, we may adopt this value for Period A, Period B, and Period C as the right-hand side of equation (7.15a). Period A In Period A, we set, from Fig. 10.17, the values F (0) = 0.04, and F (20) = 0.13. Hence, it holds that F (0) < F (20), that is, the first rolled trend score is positive. The left-hand side of equation (7.15a) is calculated as F (20) − F (0) 0.13 − 0.04 = = 0.0045. 20 20

(10.14)

From these, we have F (20) − F (0) δ(n − 1) = 0.0045 > 0.00107 = (σ(τ ))2 . 20 2

(10.15)

From Theorem 7.2.1, the first market price of risk might be positive. Period B and C For Period B, we set F (0) = 0.07 and F (20) = 0.09. From these, the rolled trend in this period is weakly positive. The same calculation as in equation (10.14) implies that F (20) − F (0) 0.09 − 0.07 = = 0.001, 20 20

(10.16)

and this is almost equal to the value from equation (10.13). Hence, Period B might imply a near-zero market price of risk. For Period C, we set F (0) = 0.13 and F (20) = 0.06. Accordingly, this period exhibits a negative rolled trend. The same calculation as in equation (10.14) implies that F (20) − F (0) 0.13 − 0.06 = = −0.0035, 20 20

(10.17)

which is obviously smaller than the value from equation (10.13). Thus, Period C might imply a negative market price of risk.

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Estimation in the Hull–White model Next, let us calculate the market price of risk for three periods by employing a one-factor Hull–White model. We calibrate the term structure for them in the manner described in Section 10.3. The results are shown in Table 10.5 for the three periods. In this, ρ is the positive square root of the eigenvalue ρ2 , σ is the coefficient for the Hull–White volatility, κ is the mean reversion rate, ϕ is the market price of risk, and R is the first rolled trend score, and β is a constant given by equation (8.25). We additionally shows the results in Section 10.3 as the last row, in which we calculate the positive market price of risk for the period from March 1975 to March 1980. The table shows that the mean reversion rate κ is positive for all periods. The first rolled trend score R is positive in Period A, and the market price of risk ϕA is positive, at 0.360, as a rough estimate from the flat yield model (see above). In Period B, the rolled trend score R is slightly negative, and the market price of risk ϕB is accordingly negative, at −0.156. Similarly, the market price of risk ϕC in Period C is negative, at −0.506. As a result, ϕA , ϕB , and ϕC take positive, negative and negative values respectively, which roughly agrees with qualitative estimates obtained by using the flat yield model. Furthermore, the highest of the market prices of risk is that of Period A, followed in descending order by that of Period B and that of Period C. Specifically, ϕA > ϕ B > ϕ C .

(10.18)

Comparing the calculations of {F (20) − F (0)}/20 in equations (10.14), (10.16) Table 10.5: Market price of risk for Periods A, B and C. The last row exhibits the result of the example in Section 10.3, which shows a positive market price of risk, for comparison. The interest rate dataset is the same as for Fig. 10.13.

Case A B C Section 10.3

Period ρ σ κ R β ϕ ’60-’80 0.033 0.011 0.089 0.0130 -0.0011 0.360 ’70-’90 0.057 0.019 0.089 -0.0056 -0.0033 -0.156 ’80-’00 0.055 0.017 0.064 -0.0245 -0.0032 -0.506 ’75-’80

0.047

0.019

0.146

0.0304

-0.0020

0.609

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Forward interest rate

0.14 0 5 10

0.12 0.1 0.08 0.06 0.04 0.02

2014/1/2

2012/1/2

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1970/1/2

0

Figure 10.18: Implied forward rates in U.S. Treasury market. The labels 0, 5, and10 indicate the forward rates for the periods [0, 0.5], [5, 5.5], and [10, 10.5] years, respectively. Comparing the calculations of {F (20) − F (0)}/20 in equations (10.14), (10.16) and (10.17) in the flat yield model, the value for Period A is the highest, followed in order by Period B and Period C, which agrees with the order in equation (10.18). In particular, both Period A and B contain the period from March 1975 to March 1980, in which the market price of risk is larger than that in Period A and B. These relations might be due to long-period observation. From these, we may take the view that the flat yield model is valid for explaining the negative price tendency. A statistical examination of this tendency would be an interesting subject for future research. 10.5

Mean Price Property

This section examines the mean price property of the market price of risk, referring to the numerical examples in Yasuoka (2017a). We use yield data for the U.S. Treasury market from 2 January 1970 to 1 January 2016. For

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Table 10.6: Specification of observation period. The first day of each period is the first Friday of the year. The period lengths (in weeks) are set such that each period is distinct. Years Weeks 1st period 2nd period 3rd period .. .

Case A 1 52 1/2/1970 1/1/1971 1/7/1972 .. .

Case B 3 156 1/2/1970 1/5/1973 1/2/1976 .. .

Last period Number of periods

1/2/2015 46

1/6/2012 15

Period length Beginning day

estimating the market price of risk for one-year periods, we use weekly data. Recall that we work with the same data used for the example in Section 10.3. We used monthly data in that section. Setting δ = 0.5 (years) and xi = δi for i = 1, 2, · · · , 20 (n = 20), the 6-month forward rate is obtained for this period. Fig. 10.18 shows a historical chart of the implied forward rates. The first market price of risk For convenience, we regard the 6-month forward rate as the instantaneous forward rate. We consider two cases, Case A and Case B. In Case A, all periods are subdivided annually, taking from the first Friday in each year to the Friday 52 weeks (one year) later. In Case B, all periods are subdivided into three-year periods, taking from the first Friday in the first year to the Friday 156 weeks (three years) later. The details are listed in Table 10.6. To calculate the market price of risk, we employ the one-factor Hull–White model. The volatility parameters κ and σ and the first market price of risk ϕ are calculated for each period according to the method in Section 8.6. Fig. 10.19 shows the chart and the histogram of the market price of risk for Cases A and B. These results are basically the same as in Yasuoka (2017a). The market price of risk takes values in the range -3.31 to 2.23 in Case A, and in the range -1.45 to 0.79 in Case B. These look to change uncertainly, as a

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10

3 Case A

CaseA

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2

8 7

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0 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014

-4

Chart of the market price of risk

0

0.5

1

1.5

2

2.5

3

Market price of risk

Histogram of the market price of risk

Figure 10.19: Chart and histogram of the first market price of risk for Cases A and B. stationary process. The histogram shows that the values of the market price of risk are distributed around zero. Table 10.7. exhibits the historical mean and standard deviation of the market price of risk for each case. From this, the market price of risk is distributed with a mean near -0.4. Table 10.7: Historical mean and standard deviation of the market price of risk. Mean Stadard deviation Period length (year) Number of periods

Case A -0.405 1.322 1 46

Case B -0.456 0.583 3 15

Positive slope model The above observations are roughly explained by the mean price property and

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the positive slope model. To see this, we roughly estimate the first market price of risk by applying the positive slope model and Theorem 7.2.3. The historical mean of the forward rates in the whole period is shown in the left panel of Fig. 10.20. The mean curve has a weakly positive slope in the short-term forward rates and is almost flat in the long term. To apply the positive slope model with our data, we use the least-squares method to fit the mean curve linearly, giving y = 0.0594 + 0.0017x. The right panel of Fig. 10.20 shows the first three volatility components for the whole period. We see that the first and second volatility components have typical shapes, as seen in Fig. 4.2. Our analysis now uses only the first volatility, which we may regard as flat in the positive slope model. The amount of volatility is roughly evaluated as σ = 0.01 from the graph. When the sample period length goes to infinity, the positive slope model implies the limit of the market price of risk by Theorem 7.2.3: 0.0017 19 × 0.5 × 0.01 − lim ϕ(τ ) = − τ →∞ 0.01 2 = −0.218. From this, we see that the first market price of risk is roughly equal to -0.22 when we work with long-period data. Furthermore, equation (7.44) roughly asserts that the mean of the market price of risk is equal to 0.22. In fact, the actual mean of the market price of risk is near -0.4 (from Table 10.7), which is roughly explained by the mean price property with the positive slope model. Mean price property To provide further information, we examine the mean price property, comparing Cases A and B. Each three-year period in Case B is divided into three annual periods in Case A. For example, we already have the market price of risk for the years 1970, 1971, and 1972 in Case A; we denote these prices by ϕ1970,1 , ϕ1971,1 , and ϕ1972,1 , respectively. Generally, we denote them by ϕt,1 for t = 1970, . . . , 2015. We have the market price of risk ϕ1970,3 for the three-year period from 1970 to 1972 in Case B. Corollary 7.3.2 claims that 1 (ϕt,1 + ϕt+1,1 + ϕt+2,1 ) = ϕt,3 3

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0.08

0.015

Mean linear Approximaon

0.075

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1st 2nd 3rd

-0.015

Volatility components

Figure 10.20: Term structure of the implied forward rates. Historical mean of the implied forward rate and linear approximation, and the volatility components. for each t = 1970, 1973, 1976, . . . , 2012, with volatility unchanged. In this respect, we calculate the average xt of the market price of risk for each of the three periods in Case A such that 1 (10.19) xt = (ϕt,1 + ϕt+1,1 + ϕt+2,1 ) 3 for t = 1970, 1973, . . . , 2012. Fig. 10.21 compares this average with ϕt,3 , where the thin line represents the average xt in Case A, and the bold line represents ϕt,3 (Case B). The graph roughly shows good correspondence between them. In a practical sense, we may have 1 (ϕt,1 + ϕt+1,1 + ϕt+2,1 ) ≈ ϕt,3 . (10.20) 3 This shows that the mean price property (10.19) is roughly valid for historical data. Additionally, it may be that 1 (ϕt,1 + ϕt+1,1 ) ≈ ϕt,2 . 2 These relations provide useful information for real-world modeling.

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Case B Average of Case A

Market price of risk

0.5

0

-0.5

-1

-1.5

-2 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 2012

Figure 10.21: Three-year average of the market price of risk in Case A and the market price of risk in Case B. 10.6

Credit Exposure Calculation

This section presents numerical examples of credit exposure calculation using the Hull–White model. The numerical results suggest the importance of using the real-world model for credit exposure measurement. For a more advanced treatment, see Yasuoka(2017b), which studies this subject under the HJM model and the Hull–White model with actual market data. 10.6.1

Interest Rate Swap

Interest rate swaps are the most widely traded type of OTC derivative based on interest rates. Assume two counterparties X and Y who contract the swap, which starts at t = 0 and ends at Tn with payment dates Ti = δi, i = 1 . . . n, where δ = 0.5 (years). At each time Ti (i > 0), X receives the 6-months LIBOR rate on a unit notional and pays to Y a fixed swap rate K on the same

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notional. We refer to X as the payer, and Y as the receiver, as in Section 1.2. In what follows, the swap value is stated in terms of the value to the receiver unless otherwise specified. The swap value is calculated in the following manner. Let B(t, Ti ) denote the price at time t of a zero-coupon bond with maturity T , and let Li denote the forward LIBOR rate over the period [Ti−1 , T1 ]. Recalling equation (1.15), the swap value at a future time t > 0 is given by n  B(t, Ti )δ(K − Li−1 ). (10.21) Vt = i=m(t)

The fixed swap rate is set such that the initial value of this contract is zero. After that, the value of the swap fluctuates according to the evolution of the interest rate. For instance, the value decreases when the swap rate increases. More precisely, the swap value has the convexity property since the bond price B(t, Ti ) has the convexity property with respect to changes in the swap rate. This is the same as the position for bondholders. On some payment day t = Tj for some j > 0, the definition of (10.21) means that the value of cash flow at Tj is neglected. The value on this day (before cash flow is accounted) is obtained by adding the unpaid δ(K − Lj−2 ) to VTj , giving V Tj − =

n 

i=j+1

10.6.2

B(t, Ti )δ(K − Li−1 ) + δ(K − Lj−2 ).

(10.22)

Numerical Conditions

As an example for calculating the exposures, the initial yield is set such that the 6-month forward LIBOR rates are a flat 3%. Accordingly, the swap rate is also equal to 3% at this time. The swap value is equal to zero at t = 0 and 10. As an interest rate scenario generator, we employ the one-factor Hull–White model, which is recalled from (8.11) as  t 2 σ −κ(T −s) f (t, T ) = f (0, T ) + e [1 − e−κ(T −s) ]ds 0 κ  t  t −κ(T −s) e ϕs ds + σ e−κ(T −s) dWs , (10.23) +σ 0

0

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0.025

Hull-White Volality 0.02

0.015

0.01

0.005

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

Figure 10.22: Hull–White volatility, with κ = 0.05 and ρ = 0.02. where the Hull–White volatility has the form σ(t, T ) = σ exp{−κ(T − t)}.

(10.24)

From equation (10.23), the simulation model is discretely represented as σ  (1 − e−κxi ) + ϕ Δs f (Δs, Ti ) = f (0, Ti ) + σe−κxi κ √ −κxi W1 . + Δsσe We set the parameters κ and ρ at 0.05 and 0.02, respectively. Fig. 10.22 shows the graph of the Hull–White volatility with these parameters. We consider Cases 1, 2, and 3, where the market price of risk is set to be 0.5, 0, and -0.5 respectively. Case 1 represents a historical period when the interest rate was rising, since the market price of risk is positive. Case 3 represents a period when the interest rate was falling. Case 2 presents to risk-neutral simulation where the savings account is taken as the numer´aire. Using a Monte Carlo simulation with 50,000 data points, we calculate the credit exposures and use these to obtain the EE and the PFE profiles of a 10-year vanilla swap for three cases.

Numerical Examples

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Figure 10.23: Mean of swap value to the receiver in Case 2. 10.6.3

Distribution of Swap Value

Observing the distribution of the swap value in Case 2 (risk-neutral simulation) as a benchmark, we examine the basic properties of credit exposure. Fig. 10.23 exhibits the mean of the swap value at each time, where “μ ± σ” represents the mean plus/minus one standard deviation. In this case, the third term in (10.23) vanishes with the condition φ = 0. The drift term consists of only the second term, which is equivalent to −σ0i · υ0i in the HJM model. Experimentally, this term takes small positive values. Each forward rate F (t, Ti ) is normally distributed with its mean slightly larger than the initial rate of 3%, and the rate gradually increases for ten years. Accordingly, the swap value decreases a little. From these, the mean of the swap value becomes more negative over the ten years, converging to zero as time approaches the tenth year. Since the standard deviation of the forward rate increases almost propor√ tionally to t, the standard deviation of the swap value increases in the early period. Several years later, the term to maturity becomes shorter, and the swap value decreases to zero. From this, the standard deviation of the swap

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0.14

2Y 8Y

0.12 0.1 0.08 0.06 0.04 0.02 0 -0.4

-0.2

0

0.2

0.4

0.6

Swap Value at Receiver

Figure 10.24: Histogram of swap value to the receiver at years 2 and 8 of Case 2. value decreases to zero near the maturity. Fig. 10.24 shows the histogram of the swap value for years two and eight. The mean, standard deviation, and skewness are listed in Table 10.8. At the second year, the skewness is strictly positive (0.511), which is caused by the convexity of the swap value, as mentioned in Section 1.4. Note the long right tail; the mass of the distribution is thus concentrated to the left of Fig. 10.24. Fig. 10.25 explains the mechanism of this property. In that figure, the horizontal axis represents the swap rate, and the vertical axis represents the swap value. The bold line in the center visualizes the swap value relative to the swap rate, which shows the property of convexity. When the swap rate rises, the receiver’s value decreases, and the distribution becomes concentrated, from the convexity. In contrast, when the swap rate falls, the receiver’s value increases, and the distribution expands to form a long tail, again from the convexity. From this, the PFE of the receiver should be larger than that of the payer in the early term of the swap contract. At the eighth year, the standard deviation is 0.068, which is markedly smaller than the standard deviation at the second year. This is explained by

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Swap Value

the sensitivity’s property, discussed in Section 1.4. Specifically, the interestrate sensitivity of the swap is much smaller at the eighth year than that at the second year. From this, the convexity does not obviously affect the distribution of the swap value; the skewness is relatively small, at 0.083. However, the mean is -0.025, which is negative and further from zero than at the second year. This is explained by the positive value of the drift term. The distribution of the swap value is approximately normally distributed with a negative mean at the eighth year. As a result, the PFE of the receiver becomes smaller than that of the payer in the later part of the profile. From these observations, it is suggested that the PFE is affected by the magnitude of the volatility in the risk-neutral model. This, of course, is affected by the choice of risk-neutral measure (e.g., spot measure or various forward measures). This means that exposure measurement lacks a sound basis when the risk-neutral model is used.

Long tail

Distribuon of swap value of receiver

Concentraon of mass

Swap value to swap rate

Distribuon of swap rate Swap rate

Figure 10.25: Convexity and skewness of swap value to the receiver

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Table 10.8: Descriptive statistics of the swap value for Case 2 Year Mean Standard deviation Skewness Kurtosis

0.3

2 −0.006 0.153 0.511 0.529

8 −0.025 0.068 0.083 0.262

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Figure 10.26: EE profiles for payer and receiver 10.6.4

Credit Exposure Calculation

EE profiles in risk-neutral simulation Fig. 10.26 shows the EE profiles of the payer and receiver for Cases 1, 2, and 3. The EE profile of the payer in Case 2 (risk-neutral simulation) is superficially similar to that of the receiver. Let us examine these profiles more precisely. As mentioned in the previous section, each forward rate F (t, Ti ) is normally distributed with its mean a little higher than the initial rate F (0, Ti ). Combining the convexity effect with this, the expectation of the receiver’s swap value Vt+ is larger than that of the

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payer’s ((−V )+ t ). That is, the receiver’s EE is larger than the EE in the early term of Fig. 10.26. In contrast, for the later period, the mean of the swap value is negative and the distribution is almost symmetric. The payer’s EE is larger than the receiver’s. Case 2 in Fig. 10.26 shows this. EE profiles in Cases1 and 3 In Case 1, the market price of risk is set to be positive, at 0.5. Accordingly, the forward rates are rising on average, and the swap value to the payer increases. Consequently, the payer’s EE profile increases, but the receiver’s profile decreases. From this, the EE profiles of the counterparties are asymmetrical. In Case 3, the market price of risk is set to be negative, at -0.5. The results are quite different from Case 1 by the same mechanism as above. Specifically, the payer’s EE profile is smaller than in Case 2, but the receiver’s profile is larger than in Case 2. PFE profiles Fig. 10.27 shows the PFE profiles of the payer and the receiver for the same three cases. In Case 2, the receiver’s profile is larger than the payer’s profile in the early term, which is mostly caused by the convexity, explained above. However, the receiver’s PFE profile is a little smaller than the payer’s profile in the later term, which is also caused by the positive drift of forward rates, as explained in Section 10.6.3. In Case 1, the PFE profiles are remarkably asymmetrical. Specifically, the payer’s PFE profile is larger than in Case 2, but the receiver’s profile is smaller than in Case 2. This is caused by the positive market price of risk and the higher interest rates. The results in Case 3 are reversed from Case 1. Real-world model and risk-neutral model From our observations, the risk-neutral model can be used to reflect the magnitude of the volatility structure in PFE measurement but cannot be used for finding the historical drift of interest rates. In contrast, the real-world model can reflect historical drift. In other words, the PFE measurement depends strongly on the choice of sample period. In this context, choice of period is important for credit exposure measurement with real-world modeling.

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Takashi Yasuoka

PFE 99% Receiver

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Figure 10.27: PFE profiles (99% confidence level) of payer and receiver.

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Appendix A BASICS OF NUMERICAL ANALYSIS This appendix notes some basic tools used for numerical analysis: normal distributions and eigenvectors. A.1

Normal Distribution

Let f (x) be a real-valued function on R such that   (x − μ)2 1 exp − . f (x) = √ 2σ 2 2πσ

(A.1)

Here, f (x) is called a normal distribution function with mean μ and variance σ 2 . In particular, when μ = 0 and σ = 1, f (x) is called the standard normal distribution. Let X be a random variable with a distribution function f (x). If f (x) is a normal distribution, then X is said to be normally distributed. For a positivevalued random variable X, if log X is normally distributed, then X is said to be lognormally distributed. Here, the distribution function of X is given by g(x) = f (log x)/x, which, written in full, is   ! (log x−μ)2 √ 1 exp − , x>0 2σ 2 2πσx g(x) = (A.2) 0 , x ≤ 0. In this form, g(x) is called a lognormal distribution function with parameters μ and σ 2 . Fig. A.1 indicates the graph of a normal distribution function with μ = 3 and σ 2 = 1 as a solid line, and a lognormal distribution function with μ = 1 and σ 2 = 0.25 is indicated by a dashed line. Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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Normal distribuon Lognormal distribuon

-1

0

1

2

3

4

5

6

7

Figure A.1: Examples of a normal distribution function and a lognormal distribution function In this book, the normal distribution appears in the distribution of interest rates in the Gaussian HJM model and again in the Hull–White model. For example, it is stated at the end of Section 4.4 that the short rate in the Hull–White model is normally distributed. Additionally, the lognormal distribution appears in the distribution of the asset price, as described in equation (2.48a). The LIBOR in the LIBOR market model, in particular, is lognormally distributed under some measures; for details, see Section 5.5. A.2

Eigenvalues and Eigenvectors

This section summarizes the basics of the eigenvalues and eigenvectors of a symmetric matrix. In addressing these, the basic results are introduced omitting the proofs. For more details, see, for example, Petersen (2012) or Simon and Blume (1994), among others. Eigenvalue Let A be an n × n real-valued matrix. If detA = 0, then we say A is invertible. In this case, the rank of A is naturally equal to n. Let x = (x1 , · · · , xn )T be an Rn -valued nonzero vector, and let λ be a scalar

Appendix

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variable. Let us consider a linear system of equations in the form Ax = λx.

(A.3)

If λ and x satisfy this equation, then we say that λ is an eigenvalue of A, and x is the eigenvector corresponding to λ. In this respect, solving the equation (A.3) is called an eigenvalue problem. Note that eigenvalues do not uniquely exist, and an eigenvalue λ may not uniquely determine an eigenvector. Indeed, if x is an eigenvector for an eigenvalue λ, then it follows for a constant c that Acx = cAx = cλx = λcx,

(A.4)

and so cx is also an eigenvector associated with the eigenvalue λ. In this book, the matrix A for the eigenvalue problem is always assumed to be a symmetric matrix. The next proposition summarizes fundamental results for the eigenvalue problem with a symmetric matrix. Proposition A.2.1 (1) A is invertible if and only if zero is not an eigenvalue. (2) If A is a symmetric real-valued matrix, then all eigenvalues of A are real, and all eigenvectors are in Rn . (3) Let A be a symmetric matrix. For two eigenvalues λ1 and λ2 of A, let x1 and x2 be eigenvectors corresponding to λ1 and λ2 , respectively. If λ1 = λ2 , then x1 and x2 are orthogonal; specifically, x1 · x2 = 0 in terms of the standard inner product in Rn . Orthogonal diagonalization Let E be an n × n real-valued matrix, and let E T denote the transpose matrix of E. If E satisfies EE T = E T E = I, then E is called an diagonal matrix. For an n × n real-valued matrix A, if there exists an orthogonal matrix E such that EAE T is a diagonal matrix (i.e., a square matrix with non-zero entries on only the main diagonal), then A is called orthogonally diagonalizable. For the diagonalization, the next proposition holds. Proposition A.2.2 If A is a symmetric matrix, then A is orthogonally diagonalizable.

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Appendix B PRINCIPAL COMPONENT ANALYSIS

This appendix briefly presents the fundamental concepts of principal component analysis (PCA) as it is used for volatility construction. B.1

Principal Component

Diagonalization Let S = {Sij } be an n × n covariance matrix derived from some sample data. When the rank of S is smaller than n, we reduce the number n of variables of the data so as to be equal to the rank. From this operation, we may assume that S is invertible. Since a covariance matrix is always symmetric, it holds from Proposition A.2.1(2) that all eigenvalues are real values, and all eigenvectors are in Rn . Furthermore, it is known that eigenvalues of covariance matrices are all nonnegative. From Proposition A.2.1(1), no eigenvalues are zero, and so they must all be positive. We assume that all eigenvalues are distinct and let (ρ1 )2 > (ρ2 )2 > · · · > (ρn )2 > 0 be eigenvalues of S and el = (el1 , · · · , eln )T be the corresponding eigenvectors. We assume that |el |2 = 1 for all l. Here (ρl )2 and el are referred to as the lth eigenvalue and the lth eigenvector, respectively. The covariance matrix S is orthogonally diagonalizable from Proposition A.2.2. Using this, we can find the orthogonal matrix that will diagonalize S. From the definition of the eigenvector, it holds that Sel = (ρ1 )2 el for 1 ≤ l ≤ n.

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Regarding (e1 , · · · , en ) as an n × n matrix, we have that S (e1 , · · · , en ) = ((ρ1 )2 e1 , · · · , (ρn )2 en ) ⎞ ⎛ (ρ1 )2 0 0 ⎜ ⎟ .. = (e1 , · · · , en ) ⎝ 0 . 0 ⎠. 0 0 (ρn )2

(B.1)

E = (e1 , · · · , en ),

(B.2)

Setting an n × n matrix E as

it follows from Proposition A.2.1(3) that all eigenvectors are pairwise orthogonal. Additionally, from the assumption that |el |2 = 1 for all l, the matrix E is an orthogonal matrix, so that EE T = E T E = I. From this and equation (B.1), we have ⎛ ⎞ (ρ1 )2 0 0 ⎜ ⎟ .. E T SE = ⎝ 0 (B.3) . 0 ⎠. 2 0 0 (ρn )

Thus, we see that an orthogonal matrix E to diagonalize S is constructed from the set of all eigenvectors.

Principal component The lth eigenvector el is also referred to as the lth principal component. In most places, this book uses the term “principal component” rather than “eigenvector.” It holds from EE T = I that n  eli ehi = δlh , 1 ≤ l, h, ≤ n. (B.4) i=1

This means that the principal components are pairwise orthogonal and each have unit length. A set of vectors collectively satisfying equation (B.4) is said to be orthonormal. B.2

Principal Component Space

State space Under the same assumptions about the covariance matrix S as in the previous

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Appendix

subsection, let e1 , · · · , en be the principal components of a covariance matrix ˜n S. Since this set is orthonormal, we can define another Euclidean space R ˜ n is sometimes referred to as a principal spanned by e1 , · · · , en . The space R component space or a factor space. In contrast, Rn is called a state space or an original space. Principal component space Let x = (x1 , · · · , xn )T be a vector in Rn . The inner product between x and el  in Rn is given by x · el = ni=1 xi eli , which is called the lth principal component score of x. Thus, the vector (xe1 , · · · , xen )T is regarded as the coordinates of ˜ n . In this context, we say that x is represented by (xe1 , · · · , xen )T in x in R the principal component space. Projection to principal component space Setting a ˜l = xel , we have ˜n )T = (xe1 , · · · , xen )T . (˜ a1 , · · · , a

(B.5)

a1 , · · · , a ˜ n )T ∈ We regard the above relation as a linear mapping from x ∈ Rn to (˜ ˜ n . Thus, the lth principal component score xel is also referred to as the lth R projection. Since E = (e1 , · · · , en ) is an orthogonal matrix, we have |x|2 = |Ex|2 = (xe1 , · · · , xen )(xe1 , · · · , xen )T n  = (˜ al ) 2 .

(B.6)

l=1

This means that the norm of (˜ a1 , · · · , a ˜n )T is equal to that of x. Inverse mapping to state space Conversely, let us consider an inverse mapping from the principal component ˜ n , equation ˜ n to the state space Rn . For an element (˜ a1 , · · · , a ˜ n )T ∈ R space R (B.5) is represented by ˜n )T = (e1 , · · · , en )T (x1 , · · · , xn )T . (˜ a1 , · · · , a

(B.7)

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Takashi Yasuoka

Left-multiplying both sides by (e1 , · · · , en ), we obtain the inverse mapping (e1 , · · · , en )(˜ a1 , · · · , a ˜n )T = (e1 , · · · , en )(e1 , · · · , en )T (x1 , · · · , xn )T = I(x1 , · · · , xn )T

= (x1 , · · · , xn )T .

(B.8)

Least squares approximation Let x be an arbitrary vector in Rn . Let us consider the least squares approximation of x by the lth principal component el . This is formulated as the problem of finding a constant a that minimizes the sum of squared differences as n  (B.9) |xi − aeli |2 . i=1

The next proposition shows that the principal component score gives the solution to this problem.

Proposition B.2.1 Let e1 , · · · , en be a set of principal components. For an arbitrary x ∈ Rn , a = xel is the unique solution that minimizes equation (B.9) for each l. This proposition is applied for the calibration in the Hull–White model in Section 8.4. B.3

Covariance and Volatility

At the end of this appendix section, we address ourselves to the volatility structure associated with the covariance matrix. Covariance matrix In the HJM model, F (tk , xi ) denotes the instantaneous forward rate with maturity tk + xi as empirically observed at tk . With this, the covariance matrix S is obtained by 1 Cov (F (tk + Δt, xi − Δt) − F (tk , xi ), Sij = Δt F (tk + Δt, xj − Δt) − F (tk , xj )) ; 1 ≤ i, j, ≤ n, (B.10)

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Appendix

as introduced in Section 4.3. From this covariance matrix, we construct a volatility as follows. As stated at the beginning of Appendix B.1, we may assume that S is invertible. Thus, because S is a covariance matrix, all eigenvalues of S are positive. We assume that all eigenvalues are distinct. Let (ρ1 )2 > (ρ2 )2 > · · · > (ρn )2 > 0 be the eigenvalues. From the principal components e1 , · · · , en , we define an orthogonal matrix (e1 , · · · , en ). It holds from equation (B.1) that ⎛ ⎞ (ρ1 )2 0 0 ⎟ 1 ⎜ n T .. Sij = (e1 , · · · , en ) ⎝ 0 . 0 ⎠ (e , · · · , e ) 0 0 (ρn )2 n  = (B.11) eli ρ2l elj . l=1

Principal volatility component We determine the volatility structure from the eigenvalues and the associated principal components. Specifically, the lth volatility component is defined by σ l = ρl el . Obviously, this volatility structure captures the covariance S. Additionally, we always assume in this book that el1 > 0,

ρl > 0 ; l = 1, · · · , n.

(B.12)

These assumptions are significant conditions that allow us to obtain the implications of the market price of risk in Sections 6.5 and 9.5. The argument here is basically the same as that for volatility construction in the LIBOR market model, where the covariance matrix is obtained by equation (9.7).

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Appendix C MAXIMUM LIKELIHOOD ESTIMATION This appendix introduces maximum likelihood estimation to find the mean of a random variable from sample data. For details of maximum likelihood estimation generally, see, for example, Cramer (1989). Likelihood function Let x1 , · · · , xn be independent samples from a random variable Y , and let us suppose that the true mean of the variable Y is unknown. Assuming that the mean of Y is equal to θ, let g(xi |θ) denote the probability of the outcome xi . Then, the joint probability of the outcome (x1 , · · · , xn ) under the condition of θ being the mean is given by g(x1 , · · · , xn |θ) =

n  i=1

g(xi |θ).

(C.1)

Our aim is to estimate the value of θ that maximizes the probability g (x1 , · · · , xn |θ). A likelihood function for a fixed (x1 , · · · , xn ) is a one-variable function of θ defined by L(θ|x1 , · · · , xn ) = g(x1 , · · · , xn |θ) n  = g(xi |θ).

(C.2)

i=1

We determine θ¯ such that θ¯ maximizes the likelihood function as ¯ 1 , · · · , xn ) = max L(θ|x1 , · · · , xn ). L(θ|x θ

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

(C.3)

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Here, the probability of outcome x1 , · · · , xn attains the maximum value when ¯ From this, we consider that the mean of Y is estimated as θ¯ because θ = θ. ¯ This method of estimating θ (x1 , · · · , xn ) is most likely to occur when θ = θ. is called maximum likelihood estimation. We note that the likelihood function is typically defined for other statistical parameters, such as the variance of Y . Log-likelihood function A log function of L(θ|x1 , · · · , xn ) is represented by log L(θ|x1 , · · · , xn ) =

n  i=1

log g(xi |θ),

(C.4)

which is called a log-likelihood function. Since the log function is strictly increasing, log L(θ|x1 , · · · , xn ) and L(θ|x1 , · · · , xn ) take the maximum value at the same value of θ. Furthermore, in terms of form, the log-likelihood function (C.4) is more convenient to work with than the likelihood function (C.2). Thus, we sometimes employ the log-likelihood function in preference to the likelihood function. Gauss–Markov’s theorem Let us additionally assume that Y has normal distribution with variance σ 2 . If the mean of Y is θ, then we see from equation (A.1) that   (x − θ)2 1 . (C.5) exp − g(x|θ) = √ 2σ 2 2πσ We have, from equations (C.4) and (C.5), that log L(θ, x1 , · · · , xn ) = =

n  i=1

n  i=1

log g(xi |θ)  n   (xi − θ)2 1 + − . log √ 2σ 2 2πσ i=1

(C.6)

This can be reduced to log L(θ, x1 , · · · , xn ) = −n log



n 1  2πσ − 2 (xi − θ)2 . 2σ i=1

(C.7)

Appendix

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The first term is a constant, and the right-hand side is negative. With this, the log-likelihood function attains its maximum when n  (xi − θ)2 (C.8) i=1

attains its minimum. Therefore, maximum likelihood estimation solves the least squares problem, equation (C.8). Then, we have the following proposition, known as the Gauss–Markov theorem.

Proposition C.0.1 Let Y and x1 , · · · , xn be as defined above. If a random variable Y has a normal distribution, then the maximum likelihood estimate of the mean θ of Y is given by the solution that minimizes equation (C.8). In practice, we sometimes need to estimate some parameters for financial data assumed to have a normal distribution. Proposition C.0.1 makes it convenient to estimate various type of parameters in practice. Time-series data As an application of Proposition C.0.1, we present a simple example. Let Xi , i = 1, · · · , J + 1 be sample data observed at times ti , i = 1, · · · , J + 1, respectively, with ti+1 − ti = ∆t. We assume that the dynamical behavior of Xi follows the equation dX = (A + Bϕ)dt + CdW.

(C.9)

Here, we suppose that A, B, and C are known constants, but ϕ is unknown. By the Euler approximation of the above, we have a discretization as √ (C.10) Xi+1 − Xi = (A + Bϕ)∆t + C ∆tW1 ; , i = 1, · · · , J, where W1 is regarded as a random variable with standard normal distribution. This equation then becomes A C Xi+1 − Xi (C.11) − =ϕ+ √ W1 ; , i = 1, · · · , J. ∆tB B ∆tB Here, the left hand-side is obtained from observation, and normally distributed with mean ϕ. According to Proposition C.0.1, the maximum likelihood estimate is the solution to the least squares problem for ϕ such that 2 J   A Xi+1 − Xi (C.12) − −ϕ . ∆tB B i=1

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Takashi Yasuoka

Since ∆t, A, and B are nonzero constants, this expression is reduced to the following least squares problem: J  i=1

{Xi+1 − Xi − (A + Bϕ)∆t}2 .

(C.13)

Similar equations appear in the proofs of Theorem 6.2.1 and Theorem 9.2.1 for the maximum likelihood estimation of the market price of risk.

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Appendix D PROOFS FOR DIMENSION REDUCTION D.1

Proof of Proposition 9.8.1

The market price of risk in the full-factor model is given by Theorem 9.3.1 as '  & i ' &  n 2  ˜i 1  |λ | ∆ K 0i ϕl = − λ0i E H eli EH κj + ρl i=1 ∆t 2 j=1 ' & & i ' n n   ˜i 1  H ∆K 1 = κj eli E λ0i E H eli − ρl i=1 ∆t ρl i=1 j=1 n n 1  k 2 l + (λ ) e ; l = 1, · · · , n. 2ρl i=1 k=1 0i i

(D.1)

We remark that Theorem 9.3.1 is also valid for a d-factor model. Then, ϕ˘l in the d-factor model is represented by ' & & i ' n n  ˜i 1  H ∆K 1  l H ϕ˘l = κj eli E λ0i E ei − ρl i=1 ∆t ρl i=1 j=1 d n 1  k 2 l + (λ ) e ; l = 1, · · · , d. 2ρl i=1 k=1 0i i

(D.2)

Let us examine the convergence1 of the three terms in the right-hand side of equation (D.2) to the respective terms in equation (D.1). We see that ˜ i /∆t] does not depend on d because this term is determined by the E H [∆K sample of the forward LIBOR. From this, the first term in equation (D.1) is independent from d. 1

Here the convergence is measured in the engineering sense of equation (9.108), as described in Section 9.8.1.

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Using this assumption, we see d (ρl )2 Cd = l=1 ≈ 1. n 2 l=1 (ρl )

(D.3)

n d 2 2 With this and from equation (9.108), we have l=1 (ρl ) . Since l=1 (ρl ) ≈ the volatility is constructed by PCA as λl0i = ρl eli , it holds that d  l=1

(λl0i )2 ≈

n 

(λl0i )2 .

(D.4)

l=1

Accordingly, the third terms in equations (D.1) and (D.2) each satisfy n n d n 1  k 2 l 1  k 2 l (λ ) e ≈ (λ ) e ; l = 1, · · · , d 2ρl i=1 k=1 0i i 2ρl i=1 k=1 0i i

(D.5)

The second term in equation (D.1) is specified from equation (9.5) as ' & i ' & i n n n     λk0j δK(tk , δj) 1 1  eli . (D.6) λ0i E H κj eli = λk0i E H ρl i=1 ρ 1 + δK(t , δj) l i=1 k j=1 j=1 k=1 Also, the second term in equation (D.2) is represented by & i ' & i ' n n  d    λk0j δK(tk , δj) 1 1  eli . (D.7) λ0i E H κj eli = λk0i E H ρl i=1 ρ 1 + δK(t , δj) l k j=1 i=1 k=1 j=1 It holds from equation (D.4) that equation (D.7) approximates equation (D.6) when Cd ≈ 1. This completes the proof. ✷ D.2

Proof of Theorem 9.8.2

We assume that we have already obtained all of the market prices of risk ϕ1 , · · · , ϕn in the full-factor model. Next, we examine the approximation of each term in the right-hand side of equation (9.110) by the corresponding term in equation (9.111) for different values of d. The first term in equation (9.111) is initially given as a constant, and so we may ignore this term for the proof. The second terms of equations (9.110) and (9.111) are represented as n  l=1

λl0i

i  δλl0j Lj (0) ∆s 1 + δLj (0) j=1

(D.8)

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Appendix

and d 

λl0i

l=1

i  δλl0j Lj (0) ∆s, 1 + δLj (0) j=1

(D.9)

respectively. It holds from the assumption and the approximation (D.4) that n 

λl0i

l=1

d i i    δλl0j Lj (0) δλl0j Lj (0) l ∆s ≈ ∆s. λ0i 1 + δLj (0) 1 + δLj (0) j=1 j=1 l=1

(D.10)

 The third terms of equations (9.110) and (9.111) are written as nl=1 (λl0i )2 ∆s/2  and dl=1 (λl0i )2 ∆s/2, respectively. For the same reason as in the above approximation, we have d n   (λl0i )2 (λl0i )2 ∆s ≈ ∆s ; i = 1, · · · , n. 2 2 l=1 l=1

(D.11)

Thus, by the same argument as above, it holds that γ˘i ≈ γi ; i = 1, · · · , n.

(D.12)

This approximation will be used at the end of this proof. For the last terms of equations (9.110) and (9.111), it follows from equation (D.4) that n 

λl0i



∆sW1l



l=1

d  l=1

√ λl0i ∆sW1l ; i = 1, · · · , n

(D.13)

in the sense of convergence in probability when Cd ≈ 1. From these, it remains to show when the fourth term in equation (9.111) approximates that in equation (9.110); specifically, d  l=1

λl0i ϕ˘l ≈

n  l=1

λl0i ϕl ; i = 1, · · · , n.

(D.14)

Here, we note that ϕ˘l depends on d for each l = 1, · · · , n. From equations (9.43), (9.51), and (9.113), it holds in the full-factor model that n  l l l γk elk λi ϕ l = ρl e i ϕ l = e i k=1

=

eli ζl

; l = 1, · · · , n.

(D.15)

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Similarly, for the d-dimensional model, equations (9.114), (9.115), and (9.116) imply that ˘ l ϕ˘l = ρl el ϕ˘l = el λ i i i

n 

γ˘k elk

k=1

=

eli ζ˘l

; l = 1, · · · , d.

(D.16)

From these, we have       n d n d            l l l  l ˘ λ0i ϕl − λ0i ϕ˘l  ≤  ei ζl  +  (ζl − ζl )ei  .        l=1

l=1

l=d+1

(D.17)

l=1

From equations (D.12), (9.113), and (9.115), we then see that ζ˘l ≈ ζl for all l ≤ d. Because of these, when |ζl | is sufficiently small for all l with d < l ≤ n, it hold that d  l=1

λl0i ϕl



n  l=1

λl0i ϕl ; i = 1, · · · , n.

This completes the proof.

(D.18) ✷

Remark In Section 10.2.2, the numerical example shows that the magnitude of the market price of risk |ϕl | is not monotonically decreasing in l. From this, the convergence liml→n λl0i ϕl = 0 holds independently from the convergence of the accumulated contribution rate as limd→n Cd = 1. Hence, the approximation (D.14) is not implied by the measurement of Cd alone.

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Duffie, D., & Kan, R. (1996). A yield-factor model of interest rates. Mathematical Finance, 6(4), 379-406. Filipovic, D. (2009). Term Structure Models, A Graduate Course. Berlin: Springer. Gatarek, D., Bachert, P., & Maksymiuk, R. (2007). The LIBOR Market Model in Practice. Chichester: John Wiley & Sons. Geiger, F. (2011). The Yield Curve and Financial Risk Premia: Implications for Monetary Policy. Berlin: Springer. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer. Gordy, M. B., & Juneja, S. (2010). Nested simulation in portfolio risk measurement. Management Science, 56(10), 1833-1848. Gupta, A. (2013). Risk Management and Simulation. Boca Raton: CRC Press. Hardy, M. R., & Wirch, J. L. (2004). The iterated CTE: A dynamic risk measure. North American Actuarial Journal, 8(4), 62-75. Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381-408. Harrison, J. M., & Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 11(3), 215-260. Heath, D., Jarrow, R., & Morton, A. (1992). Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica: Journal of the Econometric Society, 60(1), 77-105. Ho, T. S., & Lee, S. (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41(5), 1011-1029. Hull, J. C. (2000). Options, Futures, and Other Derivatives. Upper Saddle River: Prentice Hall. Hull, J. C., Sokol, A., and White, A. (2014). Short rate joint-measure models, Risk, Oct. 59-63. Hull, J. C., & White, A. D. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573-592. Hull, J. C., & White, A. D. (1994). Numerical procedures for implementing term structure models I: Single-factor models. The Journal of Derivatives, 2(1), 7-16. Hunt, P., & Kennedy, J. (2004). Financial Derivatives in Theory and Practice. Chichester: John Wiley & Sons. Ingersoll, J. E. (1987). Theory of Financial Decision Making. Totowa: Rowman & Littlefield. Jacod, J., & Protter, P. E. (2003). Probability Essentials. Berlin: Springer. James, J., & Webber, N. (2000). Interest Rate Modelling. Chichester : WileyBlackwell Publishing Ltd. Jamshidian, F. (1997). LIBOR and swap market models and measures. Finance and Stochastics, 1(4), 293-330.

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Jha, S. (2011). Interest Rate Markets: A Practical Approach to Fixed Income. Hoboken: John Wiley & Sons. Jorion, P. (2007). Value at Risk: The New Benchmark for Managing Financial Risk. New York: McGraw-Hill. Karatzas, I., & Shreve, S. E. (1998). Methods of Mathematical Finance. New York: Springer. Kriele, M., & Wolf, J. (2014). Value-oriented Risk Management of Insurance Companies. London: Springer. Litterman, R. B., & Scheinkman, J. (1991). Common factors affecting bond returns. Journal of Fixed Income, 1(1), 54-61. Longstaff, F. A., Santa-Clara, P., & Schwartz, E. S. (2002). The relative valuation of caps and swaptions: Theory and empirical evidence. Journal of Finance, 56(6), 2067-2109. Medova, E., Rietbergen, M., Villaverde, M., & Yong, Y. (2005). Modelling the longterm dynamics of the yield curve. Working Paper, Judge Business School, University of Cambridge. Mercurio, F., & Moraleda, J. M. (2000). An analytically tractable interest rate model with humped volatility. European Journal of Operational Research, 120(1), 205-214. Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183. Miltersen, K. R., Sandmann, K., & Sondermann, D. (1997). Closed form solutions for term structure derivatives with log-normal interest rates. The Journal of Finance, 52(1), 409-430. Moraleda, J. M., & Vorst, T. C. (1997). Pricing American interest rate claims with humped volatility models. Journal of Banking & Finance, 21(8), 1131-1157. Munk, C. (2011). Fixed Income Modelling. Oxford: Oxford University Press. Musiela, M., & Rutkowski, M. (1997). Continuous-time term structure models: Forward measure approach. Finance and Stochastics, 1(4), 261-291. Musiela, M., & Rutkowski, M. (2011). Martingale Methods in Financial Modelling. Berlin: Springer. Nawalkha, S. K., Soto, G. M., & Beliaeva, N. A. (2005). Interest Rate Risk Modeling: The Fixed Income Valuation Course. Hoboken: John Wiley & Sons. Nawalkha, S. K., Beliaeva, N. A., & Soto, G. M. (2007). Dynamic Term Structure Modeling: The Fixed Income Valuation Course. Hoboken: John Wiley & Sons. Nelson, C. R., & Siegel, A. F. (1987). Parsimonious modeling of yield curves. Journal of Business, 60(4), 473-489. Norman, J. (2009). Real world interest rate modelling with the BGM model. SSRN working paper, 1480174, Available from: http://papers.ssrn.com/sol3/papers.cfm?abstract id=1480174.t ¨ Oksendal, B. (2003). Stochastic Differential Equations. Berlin: Springer. Petersen, P. (2012). Linear Algebra. New York: Springer.

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Pliska, S. (1997). Introduction to Mathematical Finance. Malden: Blackwell Publishers. Poon, S., & Granger, C. W. (2003). Forecasting volatility in financial markets: A review. Journal of Economic Literature, 41(2), 478-539. Pykhtin, M. (2005). Counterparty credit risk modelling: risk management, pricing and regulation. London: Risk Books. Rao, C. R., Toutenburg, H., Shalabh, H. C., & Schomaker, M. (2008). Linear Models and Generalizations. Least Squares and Alternatives (3rd ed.). Berlin: Springer. Rebonato, R. (1996). Interest-rate Option Models. Chichester: John Wiley & Sons. Rebonato, R., Mahal, S., Joshi, M., Bucholz, L. D., & Nyholm, K. (2005). Evolving yield curves in the real-world measures: A semi-parametric approach. Journal of Risk, 7(3), 29-61. Riskmatrix. (2014). BBA LIBOR: Historical Market Rates, Available from: http://66.84.43.120/TF/bba-libor-historical-market-rates.html [Accessed 20 Dec. 2014]. Ritchken, P., & Sankarasubramanian, L. (1995). Volatility structures of forward rates and the dynamics of the term structure. Mathematical Finance, 5(1), 55-72. Ross, S. (2015). The recovery theorem. Journal of Finance, 70(2), 615-648. Sadr, A. (2009). Interest Rate Swaps and Their Derivatives: A Practitioner’s Guide. Hoboken: John Wiley & Sons. Schoenmakers, J., & Coffey, B. (1999). LIBOR rate models, related derivatives and model calibration. WIAS working paper. Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-time Models. New York: Springer. Simon, C. P., & Blume, L. (1994). Mathematics for Economists. New York: W. W. Norton & Company. Stanton, R. (1997). A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance, 52(5), 1973-2002. Tavella, D. (2003). Quantitative Methods in Derivatives Pricing: An Introduction to Computational Finance. Hoboken: John Wiley & Sons. Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177-188. Wu, L. (2009). Interest Rate Modeling: Theory and Practice. Boca Ration: CRC Press. Yamai, Y., & Yoshiba, T. (2002). On the validity of value-at-risk: Comparative analyses with expected shortfall. Monetary and Economic Studies, 20(1), 5785. Yamai, Y., & Yoshiba, T. (2005). Value-at-risk versus expected shortfall: A practical perspective. Journal of Banking & Finance, 29(4), 997-1015. Yasuoka, T. (2001). Mathematical pseudo-completion of the BGM model. International Journal of Theoretical and Applied Finance, 4(03), 375-401.

 Interest Rate Modeling for Risk Management

Takashi Yasuoka

Yasuoka, T. (2012). LIBOR market model under the real-world measure, and realworld simulation. Shibaura MOT Discussion Paper, 2012-2. Available from: http://mot-innovation.shibaura-it.ac.jp/wp-content/uploads/ 2013/07/d p 2012-02.pdf. Yasuoka, T. (2013a). LIBOR market model under the real-world measure. International Journal of Theoretical and Applied Finance, 16(04), 1350024. Yasuoka, T. (2013b). L2 -theoretical study of the relation between the LIBOR market model and the HJM model. Journal of Math-for-Industry, 5(2), 11-16. Yasuoka, T. (2015). Interest-rate simulation under the real-world measure within a Gaussian HJM framework. Quantitative Finance Letters, 3(1), 10-16. Available from: http://www.tandfonline.com/doi/full/10.1080/21649502.2014. 995213#.VNfX6-asWCk. Yasuoka, T., (2017a). Correlations between the Market Price of Interest Rate Risk and Bond Yields. Journal of Reviews on Global Economics, 6, 208-217. Yasuoka, T., (2017b). Evaluating Credit Exposure of Interest Rate Derivatives under the Real-world Measure, to appear in Journal of Risk Model Validation.

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Author Index ¨ Oksendal, B., 32, 44

Devineau, L., 25 Duffee, G. R., 86, 153 Duffie, D., 80, 170, 172

Ahn, D., 86 Alexander, C., 2 Aquilina, J., 170 Artzner, P., 25, 26

Eber, J., 25, 26

Bachert, P., 92 Basak, S., 26 Baxter, M., 32 Beliaeva, N. A., 186 Bhar, R., 185 Bielecki, T. R., 170 Bj¨ ork, T., 32, 42, 54, 80 Blume, L., 282 Brace, A., 36, 92, 97, 103, 118 Brennan, M. J., 177 Brigo, D., 92, 170, 172, 186 Cairns, A. J., 32, 54, 66, 72 Canabarro, E., 170, 172 Cesari, G., 170 Chan, K. C., 177 Charpillon, N., 170 Cheridito, P., 86, 88, 153 Chiarella, C., 185 Choudhry, M., 2 Chung, K. L., 32, 44 Cochrane, J., 86 Cox, J. C., 80, 177 Cramer, J. S., 291 De Jong, F., 86, 139, 153 Delbaen, F., 25, 26 Dempster, M. A., 87

Filipovic, D., 2, 86, 88, 153 Filipovic, Z., 170 Gao, B., 86 Gatarek, D., 32, 36, 92, 97, 103 Geiger, F., 2, 219 Glasserman, P., 2, 23 Gordy, M. B., 25 Granger, C. W., 176 Hardy, M. R., 29 Harrison, J. M., 54 Heath, D., 25, 26, 35, 65, 68, 69 Ho, T. S., 80, 81, 176 Hull, J. C., 2, 16, 32, 35, 80, 81, 112, 176 Ingersoll, E., 177 Ingersoll, J. E., 80 Jacod, J., 42 James, J., 32 Jamshidian, F., 36, 91–94, 97 Jarrow, R., 35, 65, 68, 69 Jha, S., 2 Jorion, P., 2 Juneja, S., 25 Kan, R., 80 Karatzas, I., 32, 44

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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Karolyi, G. A., 177 Kimmel, R. L., 86, 88, 153 Kreps, D. M., 54 Kriele, M., 2, 29 Lee, G., 170 Lee, S., 80, 81, 176 Litterman, R. B., 77 Loisel, S., 25 Longstaff, F. A., 177, 248 Maksymiuk, R., 92 Manda, I., 170 Medova, E. A., 87 Mercurio, F., 92, 185 Merton, R. C., 177 Miltersen, K. R., 36, 92 Moraleda, J. M., 185 Morini, M., 170, 172 Morton, A., 35, 65, 68, 69 Munk, C., 54, 66, 80, 81 Musiela, M., 36, 42, 61, 92, 97, 103, 118 Nawalkha, S. K., 2, 8, 186 Norman, J., 224 Pallavicini, A., 170, 172 Petersen, P., 282 Piazzesi, M., 86 Pliska, S. R., 32, 54, 61 Poon, S., 176 Protter, P. E., 42 Pykhtin, M., 170 Rebonato, R. , 176 Rennie, A., 32 Ritchken, P., 122 Ross, S., 112 Ross, S. A., 80, 177 Rutkowski, M., 36, 42, 61, 92, 170

Takashi Yasuoka

Sadr, A., 2, 16 Sanders, A. B., 177 Sandmann, K., 36, 92 Sankarasubramanian, L., 122 Santa-Clara, P., 139, 248 Scheinkman, J., 77 Schwartz, E. S., 177, 248 Shapiro, A., 26 Shreve, S. E., 32, 42, 44, 49, 66 Simon, C. P., 282 Sokol, A., 112 Sondermann, D., 36, 92 Soto, G. M., 186 Stanton, R., 87, 139 Tavella, D., 2, 32 To, T. D., 185 Vasicek, O., 80, 177 Villaverde, M., 87 Vorst, T. C., 185 Webber, N., 32 White, A., 112 White, A. D., 35, 80, 81, 176 Williams, R. J., 32, 44 Wirch, J. L., 29 Wolf, J., 2, 29 Wu, L., 66, 81 Yamai, Y., 26, 29 Yasuoka, T., 91, 92, 96, 111, 112, 149, 195, 196, 214, 235, 266, 267, 271 Yoshiba, T., 26, 29

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Subject Index P-Brownian motion, 49, 66, 93 P-martingale, 49 Q-Brownian motion, 49 Q-martingale, 49 σ-algebra, 32 Pk measure model, 107, 108 Ft -adapted, 37 Ft -measurable, 37 Bi num´eraire measure , 95 d-factor model, 67, 78 Absolutely continuous, 34 Accumulated contribution rate, 79, 145, 225, 241, 247 Actual measure, 33 Adapted process, 37 Annual discount rate, 3 Annual interest rate, 4 Annually compounded interest rate, 3 Annually coupon rate, 3 Arbitrage opportunity, 55 Arbitrage price, 56 Arbitrage pricing, 58, 70, 103, 144, 171, 224, 233 Arbitrage-free, 55 Arbitrage-free model, 23 Asset price equation, 47 At the money, 17 Augmented filtration, 37, 66 Backward-looking approach, 112 Basis point value, see BPV, 12 Bayes’ rule, 42

Bear, 128 Bear-flattening, 238, 247 BGM model, 92, 103 Black–Scholes model, 48 Bond, 3 Bond market, 68, 93 Bootstrapping, 8, 237 BPV, 12 Brace–Gatarek–Musiela, see BGM, 92 Brownian motion, 39, 42, 178, 196 Bull, 128 Bull-steepening, 238, 248 Butterfly trading strategy, 138 Call/put option, 17 Cap/floor option, 17 CCR, 170 Chan–Karolyi–Longstaff–Sanders, see CKLS, 87 Change of num´eraire, 61 CIR model, 87 CKLS model, 87 Coherent risk measure, 26 Complete, 93 Complete market, 63 Conditional expectation, 39 Conditional VaR, 28 Confidence level, 18, 20, 23 Constant market price of risk, 166 Continuous process, 37 Contribution rate, 79 Convexity, 10, 13, 14, 17, 274 Convexity risk, 10, 14

Takashi Yasuoka All rights reserved-© 2018 Bentham Science Publishers

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Correlation coefficient, 35 Counterparty credit risk, see CCR, 170 Covariance, 20, 35, 114, 241 Covariance matrix, 77, 199, 285, 288 Covariance process, 46 Covariance VaR model, 19 Cox–Ingersoll–Ross, see CIR, 87 Credit exposure, 171, 173, 271, 277 Credit risk, 15 Credit valuation adjustment, see CVA, 170 Cumulative density function, 35 Currency risk, 15 Currency swap, 170 Current exposure, 171 Curvature, 77 Curvature change, 78 CVA, 170

Takashi Yasuoka

Economic scenario generator, see ESG, 170 EE profile, 273, 277 Eigenvalue, 77, 186, 199, 282, 283, 285, 289 Eigenvector, 283, 285 Equivalent, 34 Equivalent martingale measure, 60, 69 ESG, 170 Euler approximation, 44, 197, 293 Euler integral, 114 Event, 33 Exchange rate risk, 15 Exotic option, 24 Expectation, 34 Expected exposure, see EE, 172 Expected shortfall, 28 Exponential martingale, 49, 74

Delta, 16 Density function, 35 Deterministic process, 44 Diagonal matrix, 283 Diffusion coefficient, 178 Diffusion term, 46 Dimension reduction, see Dimensionality reduction, 243, 255 Dimensionality reduction, 79, 224–226, 228 Discount factor, 2 Discount rate, 2 Distribution function, 35 Drift, 169 Drift coefficient, 67 Drift term, 46, 188, 222, 223, 255, 273 Duration, 12 Dynamic risk measure, 29

Factor space, 287 Filtered probability space, 37, 66 Filtration, 37 Filtration augmented, 93 First d-factor model, 78 First market price of risk, 137 Fixed rate, 6 Fixed-coupon bond, 3 Flat yield model, 153, 167, 262 Flattening, 78, 128 Floating rate, 6 Foreign exchange risk, 173 Forward LIBOR, 6, 93, 196, 271, 272 Forward measure, 33, 95, 106 Forward measure model, 95, 107 Forward rate, 4, 238 Forward rate model, 66 Forward-looking approach, 112 Full-factor model, 140, 150, 226, 228, 251 Full-factor simulation model, 223

Economic scenario, 173

Gauss–Markov’s theorem, 292

Subject Index

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Gaussian HJM model, 80, 112, 117, 129, 143, 186, 196, 247, 257 Geometric Brownian motion, 48 Girsanov’s theorem, 49, 51, 69, 105 Global financial crisis, 238 Heath–Jarrow–Morton, see HJM, 66 Hedge ratio, 16 Hessian, 121, 206 Historical drift, 89, 220 Historical measure, 33 Historical simulation model, 19, 22 Historical volatility, 177–179, 220 HJM model, 66, 95 Ho–Lee model, 80 Holding period, 18 Hull–White model, 80, 176, 257, 264, 266, 271, 272 Hull–White volatility, 181, 187, 258, 264, 272 Hump-shapedness, 77 Humped volatility, 182, 185 Implied forward LIBOR, 197, 251 Implied forward rate, 9 Implied short rate, 63 Implied volatility, 176 In the money, 17 Independent, 39 Inner step, 24 Instantaneous forward rate, 5, 66, 113 Instantaneous short rate, 67 Interest rate, 2 Interest rate futures, 16 Interest rate model, 66 Interest rate risk, 10, 15 Interest rate sensitivity, 12, 16 Interest rate shock, 169 Interest rate simulation, 169 Interest rate swap, 14, 170, 271 Inverse mapping, 287 Ito process, 46, 97

Ito’s formula, 47, 50 Jamshidian’s model, 102 Kalman filter, 88 Law of iterated expectations, 41 Least squares approximation, 288 Least squares problem, 117, 120, 182, 201, 205, 294 Level factor, 77 LIBOR, 6, 180, 196, 237, 271 LIBOR market model, 92, 93, 246 LIBOR market model under the realworld measure, see LMRW, 91 LIBOR model, see LIBOR market model, 93 Likelihood function, 291 Linear, 16 Linearity of portfolio, 16 Liquidity risk, 15 LMRW, 96, 98, 196, 229, 246, 250 Log-likelihood function, 292 Log-scale observable trend, 208 Log-scale rolled trend, 209 Lognormal distribution, 36, 281 London Interbank Offered Rate, see LIBOR, 6 Long-period observation, 157, 214, 261 Market, 54 Market measure, 33 Market price of interest rate risk, see Market price of risk, 63 Market price of risk, 61, 69, 71, 86, 96, 113, 114, 119, 128, 134, 144, 146, 153, 161, 167, 189, 193, 201, 211, 213, 230, 242, 259, 264, 266, 272 Market risk, 15 Martingale, 42 Maturity, 3 Maximum EE, 172

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Maximum expected exposure, see Maximum EE, 172 Maximum likelihood, 117, 123, 201 Maximum likelihood estimation, 292 Maximum PFE, 172 Mean, 20, 34, 291 Mean price property, 161, 268 Mean reversion, 80, 156, 178, 180, 259 Mean reversion rate, 80, 264 Measurable function, 34 Measure change, 43 Money market account, 67 Monotonicity, 27 Monte Carlo simulation, 23, 24, 88, 194, 246, 273 Monte Carlo simulation model, 19, 22 MPR score, 227, 228, 247, 248 Multi-factor model (VaR), 20 Musiela’s parametrization, 118 Natural filtration, 38 Negative market price of risk, 160 Negative price tendency, 153, 154, 157, 160, 214, 256, 261 Nelson–Siegel function, 191 Nested simulation, 24 Nonlinear asset, 23 Nonlinearity, 17 Norm-invariant condition, 182, 187 Normal distribution, 19, 35, 281, 292 Null market price of risk, 74, 105, 150, 152 Num´eraire, 57, 67 Num´eraire measure, 57 Numer´ aire, 171 Numer´ aire measure, 171 Observable trend, 124 Observable trend score, 127, 167, 210 One-factor model (VaR), 19 Original space, 287 Orthogonal matrix, 283 Orthogonally diagonalizable, 283

Takashi Yasuoka

Orthonormal, 199, 286 OTC derivatives, 170 Out of the money, 17 Outer step, 24 Over-the-counter derivatives, see OTC derivatives, 170 P&L distribution, 19 Parallel change, 77 Path-dependent, 24 Path-dependent asset, 23 Payer, 271 PCA, 77, 153, 198, 248 PCA setup, 119, 204 PFE, 172, 275 PFE profile, 273, 278 Physical measure, 33 Portfolio, 15 Positive homogeneity, 27 Positive Market price of risk, 255 Positive slope model, 159, 164, 268 Potential future exposure, see PFE, 172 Present value, 2 Pricing kernel, 55 Principal, 4 Principal component, 77, 114, 186, 199, 286, 289 Principal component analysis, see PCA, 75 Principal component score, 210, 287 Principal component space, 115, 198, 199, 287 Principal volatility component, 77, 122, 182, 199 Probability measure, 32 Probability space, 32, 37, 93 Process, 39 Profit–loss, see P&L, 18 Projection, 200, 287 Quadratic covariation process, 46

Subject Index

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Radon–Nikodym derivative, 36 Radon–Nikodym theorem, 36 Random variable, 34, 291 Random walk, 38 Real world, 33 Real-world, 172 Real-world measure, 23, 29, 33, 66 Real-world model, 74, 93, 144, 150, 186, 237, 271, 278 Real-world simulation, 139, 145, 147, 152, 173, 190, 217, 232, 244, 260 Receiver, 271 Recombining tree, 24 Relative price process, 58 Risk factor, 15 Risk management, 166 Risk measure, 15, 25 Risk-adjusted measure, 137, 138 Risk-free asset, 55 Risk-free bond, 3 Risk-free rate, 55 Risk-neutral drift, 88 Risk-neutral investor, 33 Risk-neutral measure, 33, 61, 69, 171, 178 Risk-neutral model, 74, 93, 150, 191, 276, 278 Risk-neutral pricing, 61 Risk-neutral simulation, 152, 244, 260, 273, 277 Risk-neutral valuation, 61, 71, 170 Risk-neutral world, 33 Risky asset, 55 Roll-down, 126, 135, 239 Roll-up, 126, 239 Rolled trend, 125, 213, 259 Rolled trend score, 127, 167, 210, 259, 264 Sample, 32 Sample space, 32 Savings account, 67, 80, 273

Second market price of risk, 137 Self-financing trading strategy, 54 Semi-annual interest rate, 4 Semi-annually compounded interest rate, 3 Sensitivity, 10, 14, 16, 275 Sensitivity risk, 10 Short rate, 67 Short rate distribution, 178 Short rate model, 66, 177 Short rate process, 81, 178 Simple forward rate, 4 Simple interest rate, 2 Spot LIBOR measure, 33, 105, 252 Spot LIBOR model, 103, 105, 108 Spot measure, 33 Spot measure model, 150 Standard normal distribution, 23, 114, 151, 197, 281 State price deflater, 171 State price deflator, 55, 73, 96, 144 State space, 287 State space setup, 119, 204 Steepening, 78, 128 Steepening strategy, 138 Stochastic differential equation, 46 Stochastic integral, 44, 50 Stochastic process, 37 Stock price risk, 15 Structured product, 24 Subadditivity, 27 Swap, 6 Swap rate, 6, 8, 180, 237, 271 Swap value, 271, 273 Tail conditional expectation, 28 Tail risk, 25 Term structure model, 66 Term structure of interest rates, 8 Terminal measure, 33, 106, 109 Third market price of risk, 139 Time-homogeneous model, 87

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Time-varying market price of risk, 166, 168 Trading strategy, 54 Translation invariance, 27 U.S. Treasury, 10, 76, 156, 257, 261, 266 Underlying risk factor, 16 Value at risk, see VaR, 18 VaR, 18, 84 Variance, 20, 34, 292 Vasicek model, 81 Volatility, 67, 93, 196, 268 Volatility component, 77, 182, 199, 289 Volatility estimation, 75, 176 Volatility fitting, 181 Volatility risk, 131 Wiener process, 39 Yield curve, 8 Yield to maturity, 4 Zero-coupon bond, 3, 68, 271

Takashi Yasuoka